Question 1 - LOAD DATA FROM EXTERNAL FILE

Data is given as CSV file. First two column are deleted.

mask1 <- read.csv ("C:/Users/MEHMET/Desktop/IE 582/Musk1 Dosyasının Kopyası.csv",header=FALSE, sep = ",")
mask1.active <- mask1[,3:168]

TASK 1 A - Principal Component Analysis

the variable standard deviations (the scaling is applied to each variable)

res.pca <- prcomp (mask1.active, scale = TRUE)
names(res.pca)
## [1] "sdev"     "rotation" "center"   "scale"    "x"

sdev : the standard deviations of the principal components (the square roots of the eigenvalues)

rotation : the matrix of variable loadings (columns are eigenvectors)

Eigenvalues

Variances in percentage

Cumulative variances

head(res.pca$sdev)
## [1] 7.195296 4.807327 3.556111 2.921889 2.857735 2.601871
head(unclass(res.pca$rotation)[, 1:10])
##              PC1           PC2          PC3         PC4           PC5
## V3 -0.0368698531  0.0036570678  0.101966679 -0.06953966  0.0373835724
## V4 -0.0683111424 -0.0187516841  0.170635673  0.11413651 -0.0508386602
## V5 -0.0948053725  0.0016863889  0.142775041  0.06199205 -0.1176633400
## V6  0.0925859839 -0.0001258461 -0.013430257  0.12588400  0.0005978495
## V7 -0.0009004731 -0.0023138712  0.008306548 -0.03565104  0.0152789269
## V8 -0.0609551501  0.0034399442 -0.038287645  0.04078182  0.0175225314
##             PC6         PC7         PC8           PC9         PC10
## V3 -0.037143720  0.16623414 -0.07408505  0.0318768712 -0.040546695
## V4 -0.014432198 -0.02277787  0.07358716 -0.0441729036 -0.071960330
## V5 -0.000184402 -0.05001262  0.07641516 -0.0216409021 -0.023947992
## V6  0.119116241  0.05729913 -0.01328586  0.0016978279 -0.189471983
## V7 -0.041046049  0.10712853 -0.05584153  0.0009369786  0.007787736
## V8  0.269271911  0.09787766  0.04891433  0.0531169943 -0.027194632
eig <- (res.pca$sdev)^2
variance <- eig*100/sum(eig)
cumvar <- cumsum(variance)
eig.mask1.active <- data.frame(eig = eig, variance = variance, cumvariance = cumvar)
eig.mask1.active[1:20,]
##          eig   variance cumvariance
## 1  51.772288 31.1881251    31.18813
## 2  23.110393 13.9219234    45.11005
## 3  12.645923  7.6180258    52.72807
## 4   8.537433  5.1430318    57.87111
## 5   8.166648  4.9196675    62.79077
## 6   6.769733  4.0781523    66.86893
## 7   5.388912  3.2463325    70.11526
## 8   5.050768  3.0426312    73.15789
## 9   3.306281  1.9917355    75.14963
## 10  2.850453  1.7171402    76.86677
## 11  2.571411  1.5490428    78.41581
## 12  2.376745  1.4317743    79.84758
## 13  2.207486  1.3298111    81.17739
## 14  2.108723  1.2703148    82.44771
## 15  1.812977  1.0921547    83.53986
## 16  1.657314  0.9983817    84.53824
## 17  1.437282  0.8658327    85.40408
## 18  1.355483  0.8165560    86.22063
## 19  1.256147  0.7567152    86.97735
## 20  1.185238  0.7139986    87.69135
print("13 out of 166 dimensions contain %80 of the variance")
## [1] "13 out of 166 dimensions contain %80 of the variance"

13 out of 166 dimensions contain %80 of the variance

Screen plot using base graphics Add connected line segments to the plot

barplot (eig.mask1.active [, 2], names.arg=1:nrow(eig.mask1.active), main = "Variances",xlab = "Principal Components",ylab = "Percentage of variances",
col ="steelblue")
lines (x = 1:nrow(eig.mask1.active), eig.mask1.active [, 2],type="b", pch=19, col = "red")

OR the same above with summary () function use the function summary() to extract the eigenvalues and variances from an object of class prcomp summary(res.pca)

Coordinates of individuals on the principal components

x (scores) are the coordinates of the individuals (observations) on the principal components

ind.coord <- res.pca$x
head(ind.coord[, 1:13])
##             PC1       PC2         PC3       PC4       PC5       PC6
## [1,] -0.7870220 -8.655373 -2.43062015 -8.100733 0.1408377 2.4027466
## [2,] -0.7816826 -8.535124 -2.63326679 -7.007034 1.0943637 1.0022780
## [3,] -0.2478201 -8.351507 -2.45185223 -8.097990 0.8303654 1.7224025
## [4,] -1.3549969 -9.096528 -2.63635288 -7.035515 0.4576381 1.6579415
## [5,] -1.4575178 -8.675879  0.01666451 -7.992784 2.8270748 1.5744610
## [6,] -1.4311819 -8.514454 -0.18698801 -6.926909 3.8490586 0.1900508
##             PC7       PC8        PC9        PC10       PC11       PC12
## [1,] -3.2668751 2.0698017 -1.2966531  0.18981509  0.9187314  0.1908860
## [2,] -1.9284613 3.7264700  2.0511993  0.23166496 -0.8251574 -0.8155672
## [3,] -2.3463715 2.7166908  0.2127552 -1.34376498 -1.9891208 -0.8432403
## [4,] -2.8112963 3.1577394  0.6020808  1.73326340  2.0006204  0.1807289
## [5,] -0.3365758 0.4036279 -1.7690861 -0.08671512  1.1679582 -0.3711877
## [6,]  1.0235737 2.0373243  1.5174426 -0.07555743 -0.5929691 -1.3338966
##            PC13
## [1,] -0.4398840
## [2,] -0.9262646
## [3,]  0.2025423
## [4,] -1.6270701
## [5,] -0.2937771
## [6,] -0.7169987

cos2 quality of representation for individuals on the principal components

#To calculate the cos2 of individuals, 2 simple steps are required :
# Calculate the square distance between each individual and the PCA center of gravity
# d2 = [(var1_ind_i - mean_var1)/sd_var1]^2 + …+ [(var10_ind_i - mean_var10)/sd_var10]^2 + …+..
# Calculate the cos2 = ind.coord^2/d2
# Compute the square of the distance between an individual and the
# center of gravity

center <- res.pca$center
scale<- res.pca$scale
getdistance <- function(ind_row, center, scale){
  return(sum(((ind_row-center)/scale)^2))
  }
d2 <- apply(mask1.active,1,getdistance, center, scale)

# Compute the cos2
cos2 <- function(ind.coord, d2){return(ind.coord^2/d2)}
ind.cos2 <- apply(ind.coord, 2, cos2, d2)
head(ind.cos2[, 1:13])
##               PC1       PC2          PC3       PC4          PC5
## [1,] 0.0032287931 0.3905154 3.079645e-02 0.3420701 0.0001033962
## [2,] 0.0033016069 0.3936264 3.746745e-02 0.2652973 0.0064712452
## [3,] 0.0003380671 0.3839368 3.309166e-02 0.3609811 0.0037954955
## [4,] 0.0092875230 0.4185763 3.515850e-02 0.2503889 0.0010594186
## [5,] 0.0116353975 0.4122687 1.521034e-06 0.3499046 0.0437751944
## [6,] 0.0116616132 0.4127457 1.990657e-04 0.2731794 0.0843486184
##               PC6          PC7          PC8          PC9         PC10
## [1,] 0.0300941745 0.0556328819 0.0223318127 0.0087642368 1.878140e-04
## [2,] 0.0054280125 0.0200949220 0.0750342826 0.0227342272 2.899914e-04
## [3,] 0.0163304910 0.0303056431 0.0406265912 0.0002491669 9.939776e-03
## [4,] 0.0139047028 0.0399794242 0.0504400915 0.0018337194 1.519681e-02
## [5,] 0.0135774214 0.0006204677 0.0008923101 0.0171416016 4.118538e-05
## [6,] 0.0002056403 0.0059649551 0.0236313887 0.0131097212 3.250299e-05
##             PC11         PC12         PC13
## [1,] 0.004399908 0.0001899393 0.0010086553
## [2,] 0.003679070 0.0035940492 0.0046359058
## [3,] 0.021779719 0.0039141064 0.0002258196
## [4,] 0.020246626 0.0001652261 0.0133917031
## [5,] 0.007471504 0.0007546415 0.0004727042
## [6,] 0.002001857 0.0101300889 0.0029268851

Contribution of individuals to the principal components

#The contribution of individuals (in percentage) to the principal components can be computed as follow :
#100 * (1 / number_of_individuals)*(ind.coord^2 / comp_sdev^2)
# Contributions of individuals
contrib <- function(ind.coord, comp.sdev, n.ind){
  100*(1/n.ind)*ind.coord^2/comp.sdev^2
}

ind.contrib <- t(apply(ind.coord,1, contrib,res.pca$sdev, nrow(ind.coord)))
head(ind.contrib[, 1:13])
##               PC1       PC2          PC3      PC4          PC5         PC6
## [1,] 0.0025134452 0.6810160 9.814693e-02 1.614784 0.0005102551 0.179158517
## [2,] 0.0024794570 0.6622248 1.151946e-01 1.208187 0.0308086446 0.031174384
## [3,] 0.0002492118 0.6340382 9.986909e-02 1.613690 0.0177373083 0.092064203
## [4,] 0.0074502749 0.7522064 1.154648e-01 1.218029 0.0053875773 0.085302142
## [5,] 0.0086203211 0.6842466 4.613472e-06 1.572034 0.2056003328 0.076928159
## [6,] 0.0083116144 0.6590212 5.808581e-04 1.180714 0.3811169308 0.001120884
##             PC7         PC8         PC9         PC10       PC11
## [1,] 0.41606101 0.178194017 0.106831886 0.0026554655 0.06896022
## [2,] 0.14498191 0.577604949 0.267343122 0.0039554866 0.05562821
## [3,] 0.21462756 0.306984223 0.002876165 0.1330839979 0.32325395
## [4,] 0.30810958 0.414751735 0.023033656 0.2214156607 0.32700237
## [5,] 0.00441629 0.006776384 0.198861730 0.0005542029 0.11144898
## [6,] 0.04084418 0.172645772 0.146311279 0.0004207591 0.02872669
##             PC12        PC13
## [1,] 0.003220762 0.018414981
## [2,] 0.058793585 0.081651682
## [3,] 0.062851125 0.003904152
## [4,] 0.002887126 0.251946031
## [5,] 0.012178606 0.008213551
## [6,] 0.157273199 0.048925102
# Note that the sum of all the contributions per column is 100

Graph of individuals: base graph

ind.coord <- cbind(ind.coord, mask1$V1)
colnames(ind.coord)[colnames(ind.coord)==""] <- "V1"

plot(ind.coord[,1], ind.coord[,2], col = ind.coord[,167]+ 1 , pch = 19,xlab="PC1 – 31.18%",ylab="PC2 – 13.92%")
legend("bottomright",cex = 0.55, legend = c("1", "0"), fill = 1:2)

abline(h=0, v=0, lty = 2)
text(ind.coord[,1], ind.coord[,2], labels=rownames(ind.coord),cex=0.7, pos = 3)

TASK 1 A - Multidimensional Scaling

Data is given as CSV file. First two column are deleted.

# EXTRACT FİRST TO COLUMNS
mask2.active <- mask1 [,3:168]
#head (mask2.active)

euclidean distances between the rows with scaling

distance_part_1a <- dist(scale(mask2.active))
distance_part_1a <-data.matrix(distance_part_1a)
fit <- cmdscale(distance_part_1a,eig=TRUE, k=166)

k is the number of dim fit view results plot solution

x <- fit$points[,1]
y <- fit$points[,2]
plot(x, y, xlab="Coordinate 1", ylab="Coordinate 2",col= mask1[,1]+1, main="Metric MDS")
abline(h=0, v=0, lty = 2)

#text(x, y, labels = row.names(mask2.active), col=mask1[,1]+1, cex=.7)

Comparing MDS and PCA

Mathematically and conceptually, there are close correspondences between MDS and other methods used to reduce the dimensionality of complex data, such as Principal components analysis (PCA) and factor analysis. PCA is more focused on the dimensions themselves, and seek to maximize explained variance, whereas MDS is more focused on relations among the #scaled objects. MDS projects n-dimensional data points to a (commonly) 2-dimensional space such that similar objects in the n-dimensional space will be

Conclusion

It is expected that BAG classes 0 and 1 (in graph black reds) has to be seperated in in different areas of the graphs. But in the graphs, this is not observed The explation is : the graphs are two dimensional these two dimensions do not cover 80 % of the variance (or similarities)

TASK 1 B - Principal Component Analysis

to take the average of multiple rows

mask1.average <- aggregate(mask1[, -c(1:2)], by = list(mask1$V1,mask1$V2),mean, na.rm  = TRUE)
res.pca_1b <- prcomp (mask1.average, scale = TRUE)
names(res.pca_1b)
## [1] "sdev"     "rotation" "center"   "scale"    "x"
head(res.pca_1b$sdev)
## [1] 6.660941 5.626150 4.751073 4.207714 3.101553 2.760043
head(unclass(res.pca_1b$rotation)[, 1:10])
##                 PC1          PC2         PC3          PC4         PC5
## Group.1  0.04463122 -0.003076793  0.11113592  0.019872124  0.02056881
## Group.2 -0.02413009 -0.011928212 -0.11958828 -0.014906750 -0.01862627
## V3      -0.05321657 -0.012972472  0.02478796  0.073524141  0.06964920
## V4      -0.06553578  0.011768063 -0.06736197  0.184556038 -0.02752374
## V5      -0.06083946  0.001440391 -0.05051918  0.187736017 -0.05825904
## V6       0.08390726  0.049635022 -0.09584341 -0.009168136 -0.17672207
##                 PC6         PC7         PC8         PC9         PC10
## Group.1  0.06914295  0.06538766  0.15233372  0.06730400 -0.093554000
## Group.2 -0.04831986 -0.10181044 -0.15128833 -0.14453079  0.076071158
## V3      -0.08866391 -0.05048083  0.17430128 -0.00711550  0.208376377
## V4       0.04187090 -0.05216477  0.02232890  0.00931202 -0.015632790
## V5       0.10236701 -0.00873377  0.01885503 -0.04989228 -0.006073795
## V6      -0.04027436 -0.03488311  0.09690788 -0.02224686  0.044453511

Eigenvalues

Variances in percentage

Cumulative variances

eig_1b <- (res.pca_1b$sdev)^2
variance_1b <- eig_1b*100/sum(eig)
cumvar_1b <- cumsum(variance_1b)
eig.mask1.average <- data.frame(eig_1b = eig_1b, variance_1b = variance_1b, cumvariance_1b = cumvar_1b)
head(eig.mask1.average)
##      eig_1b variance_1b cumvariance_1b
## 1 44.368129    26.72779       26.72779
## 2 31.653567    19.06841       45.79620
## 3 22.572693    13.59801       59.39421
## 4 17.704854    10.66557       70.05979
## 5  9.619633     5.79496       75.85474
## 6  7.617840     4.58906       80.44380

6 out of 166 dimensions contain %80 of the variance

OR the same above with summary () function

summary(res.pca_1b)

Screen plot using base graphics :

barplot(eig.mask1.average [, 2], names.arg=1:nrow(eig.mask1.average), 
       main = "Variances",
       xlab = "Principal Components",
       ylab = "Percentage of variances",
       col ="steelblue")

# Add connected line segments to the plot

lines (x = 1:nrow(eig.mask1.average),eig.mask1.average [, 2],type="b", pch=19, col = "red")

print("6 out of 166 dimensions contain %80 of the variance ")
## [1] "6 out of 166 dimensions contain %80 of the variance "

Coordinates of individuals on the principal components

ind.coord_1b <- res.pca_1b$x
head(ind.coord_1b[, 1:6])
##            PC1      PC2      PC3        PC4         PC5         PC6
## [1,] -1.419314 7.394696 9.563691 -3.5113875 -0.01786291  4.08463393
## [2,] -3.361438 6.796293 9.943455 -1.9836347  1.39503368  0.08618399
## [3,] -4.243942 8.410518 7.984341 -0.1969055  0.03572679 -1.05800465
## [4,] -3.238588 8.823437 7.869699 -1.3828248 -1.23653361  4.81568481
## [5,] -5.108463 8.266105 7.620072  1.8165003  0.22273292 -5.07839888
## [6,] -5.856240 8.285880 6.978276 -1.1532488  2.05005540 -1.17789410

Quality of representation for individuals on the principal components

Compute the square of the distance between an individual and the center of gravity

center_1b <- res.pca_1b$center
scale_1b <- res.pca_1b$scale
getdistance_1b <- function(ind_row_1b, center_1b, scale_1b){
  return(sum(((ind_row_1b-center_1b)/scale_1b)^2))
  }
d2_1b <- apply(mask1.average,1,getdistance_1b, center_1b, scale_1b)

Compute the cos2

cos2_1b <- function(ind.coord_1b, d2){return(ind.coord_1b^2/d2_1b)}
ind.cos2_1b <- apply(ind.coord_1b, 2, cos2_1b, d2_1b)
head(ind.cos2_1b[, 1:6])
##             PC1       PC2       PC3          PC4          PC5          PC6
## [1,] 0.01043073 0.2831379 0.4735965 0.0638432403 1.652197e-06 8.639004e-02
## [2,] 0.06242652 0.2551896 0.5462525 0.0217391337 1.075197e-02 4.103666e-05
## [3,] 0.10142213 0.3983266 0.3589814 0.0002183281 7.187567e-06 6.303315e-03
## [4,] 0.05605824 0.4161057 0.3310125 0.0102202708 8.172217e-03 1.239493e-01
## [5,] 0.11077036 0.2900313 0.2464684 0.0140059977 2.105775e-04 1.094704e-01
## [6,] 0.18485244 0.3700537 0.2624725 0.0071685862 2.265263e-02 7.478250e-03

Contribution of individuals to the principal components

Contributions of individuals

contrib_1b <- function(ind.coord_1b, comp.sdev_1b, n.ind_1b){
  100*(1/n.ind_1b)*ind.coord_1b^2/comp.sdev_1b^2
}

ind.contrib_1b <- t(apply(ind.coord_1b,1, contrib_1b, res.pca_1b$sdev, nrow(ind.coord_1b)))
head(ind.contrib_1b[, 1:6])
##             PC1      PC2      PC3         PC4          PC5         PC6
## [1,] 0.04935125 1.877717 4.404330 0.756967670 3.605439e-05 2.380601021
## [2,] 0.27681609 1.586111 4.761057 0.241570123 2.198989e-01 0.001059823
## [3,] 0.44124514 2.429042 3.069778 0.002380322 1.442254e-04 0.159718597
## [4,] 0.25695227 2.673407 2.982257 0.117396231 1.727689e-01 3.308997535
## [5,] 0.63932473 2.346342 2.796063 0.202577292 5.605605e-03 3.679882521
## [6,] 0.84019245 2.357582 2.344904 0.081651815 4.748811e-01 0.197967012

Graph of individuals: base graph

ind.coord_1b <- cbind(ind.coord_1b, mask1.average$Group.1)
colnames(ind.coord_1b)[colnames(ind.coord_1b)==""] <- "Group.1"

plot(ind.coord_1b[,1], ind.coord_1b[,2], col = ind.coord_1b[,93]+1 , pch = 19,  
     xlab="PC1 – 26.41%",ylab="PC2 – 18.84%")
legend("bottomright",cex = 0.55, legend = c("1", "0"), fill = 1:2)
abline(h=0, v=0, lty = 2)
text(ind.coord_1b[,1], ind.coord_1b[,2], labels=rownames(ind.coord_1b),
        cex=0.7, pos = 3)

TASK 1 B - Multidimensional Scaling

EXTRACT FIRST TO COLUMNS

mask1.average_1b <- aggregate(mask1[, -c(1:2)], by = list(mask1$V1,mask1$V2),mean, na.rm  = TRUE)

euclidean distances between the rows with scaling

distance_part_1b <- dist(scale(mask1.average_1b))
distance_part_1b <-data.matrix(distance_part_1b)
fit_1b <- cmdscale(distance_part_1b,eig=TRUE, k=91)

k is the number of dim fit view results plot solution

x <- fit_1b$points[,1]
y <- fit_1b$points[,2]
plot(x, y, xlab="Coordinate 1", ylab="Coordinate 2", col= mask1.average_1b[,1]+1, main="Metric MDS Average")
abline(h=0, v=0, lty = 2)

# text(x, y, labels = row.names(mask1.average_1b), col= mask1.average_1b[,1]+1, cex=.7)

Comparing MDS and PCA

Mathematically and conceptually, there are close correspondences between MDS and other methods used to reduce the dimensionality of complex data, such as Principal components analysis (PCA) and factor analysis. PCA is more focused on the dimensions themselves, and seek to maximize explained variance, whereas MDS is more focused on relations among the #scaled objects. MDS projects n-dimensional data points to a (commonly) 2-dimensional space such that similar objects in the n-dimensional space will be

Conclusion

It is expected that BAG classes 0 and 1 (in graph black reds) has to be seperated in in different areas of the graphs. But in the graphs, this is not observed The explation is : the graphs are two dimensional these two dimensions do not cover 80 % of the variance (or similarities)

TASK 1 C – BONUS Can you think about any other # way to represent each bag with a vector?

Kernel method could be an alternative distance measurement for dissimilarity. Probablity it will overcome the problems in part a and b

Question 2 - LOAD DATA FROM EXTERNAL FILE

A Turkish Airlines plane picture is selected.

Load the neccassery libraries.

library(imager)
## Loading required package: magrittr
## 
## Attaching package: 'imager'
## The following object is masked from 'package:magrittr':
## 
##     add
## The following objects are masked from 'package:stats':
## 
##     convolve, spectrum
## The following object is masked from 'package:graphics':
## 
##     frame
## The following object is masked from 'package:base':
## 
##     save.image
require(jpeg)
## Loading required package: jpeg
require(imager)
require(data.table)
## Loading required package: data.table

TASK 2-1 Read image, display it, investigate dimensions

The picture is resized to 256 x 256 using Microsoft Paint

image_plane <-load.image("C:/Users/MEHMET/Desktop/THY_7773ER_lowres_256.jpg")
str(image_plane)
##  'cimg' num [1:256, 1:256, 1, 1:3] 0.953 0.945 0.957 0.933 0.922 ...
plot(image_plane)

dim(image_plane)
## [1] 256 256   1   3
#display(image_plane)
class(image_plane)
## [1] "cimg"         "imager_array" "numeric"
typeof(image_plane)
## [1] "double"
range(image_plane)
## [1] 0 1
hist(image_plane)

print(image_plane)
## Image. Width: 256 pix Height: 256 pix Depth: 1 Colour channels: 3

Displaying Image

Let’s display the image. RasterImage function is usede.

par(mfrow = c(1,1))
x <- 1:256
y <- 1:256
plot(x,y, ann = FALSE, axes = FALSE, col = 0)
rasterImage(image_plane, 0,0,256,256)

# TASK 2-1 EXTRA display each channel separately using ‘’image‘’ function on a single plot

display each channel seperately 3 matrices are created seperately for each channel

red <- t(apply(image_plane[,,1],2,rev))
green <- t(apply(image_plane[,,2],2,rev))
blue <- t(apply(image_plane[,,3],2,rev))

Color palettes are created as follows.

red_palette <- colorRampPalette(c("black","red"))
blue_palette <- colorRampPalette(c("black","blue"))
green_palette <- colorRampPalette(c("black","green")) 

Then, it is displayed below.

par(mfrow = c(1,3))
image(red, col = red_palette(256), ann = TRUE, axes = FALSE, main = "RED Channel" )
image(green, col = green_palette(256), ann = TRUE, axes = FALSE, main = "GREEN Channel")
image(blue, col = blue_palette(256), ann = TRUE, axes = FALSE, main = "BLUE Channel" )

TASK 2-2 - A noisy image

Noise is added to the picture. First, n oise is added each of the channels. Then, They are normalized, and at finally the noisy image is displayed side-by-side with the original image.

noise_red <-  matrix(runif(65536, min = 0, max = 2.56),256)
noise_green <-  matrix(runif(65536, min = 0, max = 2.56),256)
noise_blue <-  matrix(runif(65536, min = 0, max = 2.56),256)

image_plane_noise <- image_plane

image_plane_noise[,,1] <- image_plane[,,1] + noise_red
image_plane_noise [,,2] <- image_plane [,,2] + noise_green
image_plane_noise [,,3] <- image_plane [,,3] + noise_blue

image_plane_noise_red<- t(apply(image_plane_noise[,,1],2,rev))
image_plane_noise_green <- t(apply(image_plane_noise[,,2],2,rev))
image_plane_noise_blue <- t(apply(image_plane_noise[,,3],2,rev))

par(mfrow = c(1,2))

plot(x,y, axes = FALSE, col = 0, xlab = "",ylab = "", main = "image_plane_noise")
rasterImage(image_plane_noise, 0,0,256,256)
plot(x,y, axes = FALSE ,xlab = "",ylab = "", col = 0, main = "image_plane ")
rasterImage(image_plane, 0,0,256,256)

TASK 2-2 – B display each channel separately using ‘’image ‘’ function on asingle plot

The channel’s of the noisy image are displayed side-by-side

red_noise <- t(apply(image_plane_noise[,,1],2,rev))
green_noise <- t(apply(image_plane_noise[,,2],2,rev))
blue_noise <- t(apply(image_plane_noise[,,3],2,rev))

red_palette_noise <- colorRampPalette(c("black","red"))
green_palette_noise <- colorRampPalette(c("black","green"))
blue_palette_noise <- colorRampPalette(c("black","blue"))

par(mfrow = c(1,3))
image(red_noise, col = red_palette_noise(256), ann = TRUE, axes = FALSE, main = "RED Channel" , xlab = "",ylab = "")
image(green_noise, col = green_palette_noise(256), ann = TRUE, axes = FALSE, main = "GREEN Channel" , xlab = "",ylab = "")
image(blue_noise, col = blue_palette_noise(256), ann = TRUE, axes = FALSE, main = "BLUE Channel", xlab = "",ylab = "" )

TASK 3 a noisy image to gray scale

Transformation the image to greyscale. I will use a straightforward approach to do that. The averages of each channel are taken and normalized. Then, the greyscale image is displayed.

image_plane_noise_gray <- image_plane_noise[,,1] + image_plane_noise[,,2] + image_plane_noise[,,3]
image_plane_noise_gray <- image_plane_noise_gray / max(image_plane_noise_gray)
par(mfrow = c(1,1))
plot(x,y, main = "Greyscale Image", axes = FALSE, col = 0)
rasterImage(image_plane_noise_gray,0,0,256,256)

The next step is the PCA analysis. The 25x25 channels are used in the analysis. There will be 232*232 patches.

The PCA analysis, a feature vector is udsed. The feature vector will be the vector of patches. The 25x25 patches will be transformed into vectors with length 625, for each of 232x232 patches.

feature_vector <- rep(NA,625)

image_plane_noise_gray_data <- rep(list(feature_vector), 208*208)
k <- 1

for (i in 13:244) {
  for (j in 13:244) {
   image_plane_noise_gray_data[[k]] <- as.vector(image_plane_noise_gray[(i-12):(i+12),(j-12):(j+12)])
    k <- k+1
  }
}
image_plane_noise_gray_data <- as.data.table(matrix(unlist(image_plane_noise_gray_data), ncol = 625, byrow = TRUE))

The data is with dimensions (232x232)x625. The PCA analysis is started.

image_plane_noise_gray_PCA<- princomp (image_plane_noise_gray_data)
plot(image_plane_noise_gray_PCA)

summary(image_plane_noise_gray_PCA)
## Importance of components:
##                            Comp.1     Comp.2     Comp.3      Comp.4
## Standard deviation     0.86308239 0.38877170 0.35019317 0.277397709
## Proportion of Variance 0.06694861 0.01358396 0.01102180 0.006915806
## Cumulative Proportion  0.06694861 0.08053257 0.09155437 0.098470178
##                            Comp.5      Comp.6      Comp.7     Comp.8
## Standard deviation     0.24004895 0.222037730 0.217071298 0.18617122
## Proportion of Variance 0.00517889 0.004430887 0.004234888 0.00311503
## Cumulative Proportion  0.10364907 0.108079955 0.112314842 0.11542987
##                             Comp.9    Comp.10    Comp.11     Comp.12
## Standard deviation     0.185698425 0.17272653 0.15945525 0.157110682
## Proportion of Variance 0.003099228 0.00268136 0.00228515 0.002218444
## Cumulative Proportion  0.118529100 0.12121046 0.12349561 0.125714055
##                           Comp.13     Comp.14    Comp.15     Comp.16
## Standard deviation     0.15505641 0.148699209 0.14793969 0.145660885
## Proportion of Variance 0.00216081 0.001987259 0.00196701 0.001906878
## Cumulative Proportion  0.12787486 0.129862123 0.13182913 0.133736011
##                            Comp.17     Comp.18     Comp.19    Comp.20
## Standard deviation     0.144544503 0.140890794 0.140860465 0.14015398
## Proportion of Variance 0.001877761 0.001784031 0.001783263 0.00176542
## Cumulative Proportion  0.135613772 0.137397802 0.139181065 0.14094648
##                            Comp.21   Comp.22     Comp.23    Comp.24
## Standard deviation     0.139575894 0.1393730 0.139319797 0.13921006
## Proportion of Variance 0.001750886 0.0017458 0.001744467 0.00174172
## Cumulative Proportion  0.142697371 0.1444432 0.146187637 0.14792936
##                           Comp.25     Comp.26    Comp.27    Comp.28
## Standard deviation     0.13911291 0.139042170 0.13894968 0.13888359
## Proportion of Variance 0.00173929 0.001737521 0.00173521 0.00173356
## Cumulative Proportion  0.14966865 0.151406169 0.15314138 0.15487494
##                            Comp.29     Comp.30     Comp.31     Comp.32
## Standard deviation     0.138725987 0.138649110 0.138549373 0.138405108
## Proportion of Variance 0.001729628 0.001727712 0.001725227 0.001721636
## Cumulative Proportion  0.156604567 0.158332279 0.160057506 0.161779142
##                           Comp.33    Comp.34    Comp.35     Comp.36
## Standard deviation     0.13837713 0.13830516 0.13817555 0.137998673
## Proportion of Variance 0.00172094 0.00171915 0.00171593 0.001711539
## Cumulative Proportion  0.16350008 0.16521923 0.16693516 0.168646701
##                          Comp.37     Comp.38     Comp.39     Comp.40
## Standard deviation     0.1378196 0.137701568 0.137683737 0.137594853
## Proportion of Variance 0.0017071 0.001704178 0.001703736 0.001701537
## Cumulative Proportion  0.1703538 0.172057978 0.173761715 0.175463252
##                           Comp.41     Comp.42     Comp.43     Comp.44
## Standard deviation     0.13754846 0.137456426 0.137359536 0.137269636
## Proportion of Variance 0.00170039 0.001698115 0.001695722 0.001693503
## Cumulative Proportion  0.17716364 0.178861757 0.180557479 0.182250983
##                            Comp.45    Comp.46     Comp.47    Comp.48
## Standard deviation     0.136883575 0.13684655 0.136697222 0.13654946
## Proportion of Variance 0.001683991 0.00168308 0.001679409 0.00167578
## Cumulative Proportion  0.183934974 0.18561805 0.187297463 0.18897324
##                           Comp.49     Comp.50     Comp.51     Comp.52
## Standard deviation     0.13652663 0.136502695 0.136360493 0.136333318
## Proportion of Variance 0.00167522 0.001674633 0.001671145 0.001670479
## Cumulative Proportion  0.19064846 0.192323095 0.193994241 0.195664720
##                           Comp.53     Comp.54     Comp.55     Comp.56
## Standard deviation     0.13625864 0.136109159 0.135850941 0.135822011
## Proportion of Variance 0.00166865 0.001664991 0.001658679 0.001657973
## Cumulative Proportion  0.19733337 0.198998360 0.200657039 0.202315012
##                           Comp.57     Comp.58     Comp.59     Comp.60
## Standard deviation     0.13576372 0.135686634 0.135539307 0.135457213
## Proportion of Variance 0.00165655 0.001654669 0.001651078 0.001649079
## Cumulative Proportion  0.20397156 0.205626232 0.207277310 0.208926388
##                            Comp.61     Comp.62     Comp.63     Comp.64
## Standard deviation     0.135324749 0.135269335 0.135183559 0.135127037
## Proportion of Variance 0.001645855 0.001644507 0.001642422 0.001641049
## Cumulative Proportion  0.210572243 0.212216750 0.213859173 0.215500222
##                            Comp.65     Comp.66     Comp.67     Comp.68
## Standard deviation     0.134981404 0.134956419 0.134789315 0.134717784
## Proportion of Variance 0.001637514 0.001636908 0.001632856 0.001631124
## Cumulative Proportion  0.217137736 0.218774643 0.220407500 0.222038624
##                            Comp.69     Comp.70     Comp.71     Comp.72
## Standard deviation     0.134652873 0.134592300 0.134550832 0.134502372
## Proportion of Variance 0.001629552 0.001628087 0.001627084 0.001625912
## Cumulative Proportion  0.223668176 0.225296263 0.226923346 0.228549258
##                            Comp.73    Comp.74     Comp.75    Comp.76
## Standard deviation     0.134469994 0.13434422 0.134289386 0.13423526
## Proportion of Variance 0.001625129 0.00162209 0.001620767 0.00161946
## Cumulative Proportion  0.230174387 0.23179648 0.233417244 0.23503670
##                            Comp.77     Comp.78     Comp.79     Comp.80
## Standard deviation     0.134211827 0.134147023 0.134114820 0.134049754
## Proportion of Variance 0.001618895 0.001617332 0.001616556 0.001614987
## Cumulative Proportion  0.236655599 0.238272931 0.239889487 0.241504474
##                            Comp.81     Comp.82    Comp.83     Comp.84
## Standard deviation     0.134032095 0.133975098 0.13390578 0.133888923
## Proportion of Variance 0.001614562 0.001613189 0.00161152 0.001611114
## Cumulative Proportion  0.243119036 0.244732225 0.24634375 0.247954859
##                            Comp.85     Comp.86     Comp.87     Comp.88
## Standard deviation     0.133830484 0.133810466 0.133774530 0.133750598
## Proportion of Variance 0.001609708 0.001609227 0.001608363 0.001607787
## Cumulative Proportion  0.249564568 0.251173794 0.252782157 0.254389944
##                            Comp.89     Comp.90    Comp.91     Comp.92
## Standard deviation     0.133635890 0.133559560 0.13353091 0.133513714
## Proportion of Variance 0.001605031 0.001603198 0.00160251 0.001602097
## Cumulative Proportion  0.255994975 0.257598172 0.25920068 0.260802779
##                            Comp.93     Comp.94     Comp.95     Comp.96
## Standard deviation     0.133453250 0.133410546 0.133318138 0.133294178
## Proportion of Variance 0.001600646 0.001599622 0.001597407 0.001596833
## Cumulative Proportion  0.262403426 0.264003048 0.265600455 0.267197287
##                            Comp.97     Comp.98     Comp.99    Comp.100
## Standard deviation     0.133225818 0.133204041 0.133173412 0.133124195
## Proportion of Variance 0.001595195 0.001594674 0.001593941 0.001592763
## Cumulative Proportion  0.268792483 0.270387157 0.271981097 0.273573860
##                           Comp.101    Comp.102    Comp.103    Comp.104
## Standard deviation     0.133067404 0.133026839 0.132995856 0.132965018
## Proportion of Variance 0.001591404 0.001590434 0.001589693 0.001588956
## Cumulative Proportion  0.275165264 0.276755698 0.278345391 0.279934347
##                           Comp.105    Comp.106    Comp.107    Comp.108
## Standard deviation     0.132915110 0.132860853 0.132841611 0.132820926
## Proportion of Variance 0.001587763 0.001586467 0.001586008 0.001585514
## Cumulative Proportion  0.281522111 0.283108578 0.284694586 0.286280100
##                           Comp.109    Comp.110    Comp.111    Comp.112
## Standard deviation     0.132769297 0.132757330 0.132728215 0.132645211
## Proportion of Variance 0.001584282 0.001583996 0.001583301 0.001581322
## Cumulative Proportion  0.287864382 0.289448378 0.291031679 0.292613001
##                           Comp.113    Comp.114    Comp.115    Comp.116
## Standard deviation     0.132602912 0.132573348 0.132536359 0.132511072
## Proportion of Variance 0.001580313 0.001579609 0.001578727 0.001578125
## Cumulative Proportion  0.294193314 0.295772923 0.297351650 0.298929775
##                           Comp.117    Comp.118    Comp.119    Comp.120
## Standard deviation     0.132458227 0.132358902 0.132321288 0.132297576
## Proportion of Variance 0.001576867 0.001574503 0.001573608 0.001573044
## Cumulative Proportion  0.300506642 0.302081145 0.303654752 0.305227796
##                           Comp.121   Comp.122    Comp.123    Comp.124
## Standard deviation     0.132274311 0.13220445 0.132116490 0.132071606
## Proportion of Variance 0.001572491 0.00157083 0.001568741 0.001567675
## Cumulative Proportion  0.306800287 0.30837112 0.309939858 0.311507533
##                           Comp.125    Comp.126    Comp.127    Comp.128
## Standard deviation     0.132013038 0.132008839 0.131937324 0.131901026
## Proportion of Variance 0.001566285 0.001566185 0.001564489 0.001563628
## Cumulative Proportion  0.313073818 0.314640003 0.316204491 0.317768119
##                           Comp.129    Comp.130    Comp.131    Comp.132
## Standard deviation     0.131834234 0.131812250 0.131808442 0.131786051
## Proportion of Variance 0.001562045 0.001561524 0.001561434 0.001560903
## Cumulative Proportion  0.319330164 0.320891688 0.322453122 0.324014025
##                          Comp.133    Comp.134    Comp.135   Comp.136
## Standard deviation     0.13175087 0.131699141 0.131672693 0.13163429
## Proportion of Variance 0.00156007 0.001558845 0.001558219 0.00155731
## Cumulative Proportion  0.32557409 0.327132940 0.328691159 0.33024847
##                         Comp.137    Comp.138    Comp.139    Comp.140
## Standard deviation     0.1316170 0.131591774 0.131542634 0.131519502
## Proportion of Variance 0.0015569 0.001556304 0.001555142 0.001554595
## Cumulative Proportion  0.3318054 0.333361674 0.334916817 0.336471412
##                           Comp.141    Comp.142    Comp.143    Comp.144
## Standard deviation     0.131464054 0.131462456 0.131416824 0.131412685
## Proportion of Variance 0.001553285 0.001553247 0.001552169 0.001552071
## Cumulative Proportion  0.338024697 0.339577944 0.341130113 0.342682184
##                          Comp.145    Comp.146    Comp.147    Comp.148
## Standard deviation     0.13135294 0.131319187 0.131290905 0.131270389
## Proportion of Variance 0.00155066 0.001549864 0.001549196 0.001548712
## Cumulative Proportion  0.34423284 0.345782708 0.347331904 0.348880616
##                           Comp.149    Comp.150    Comp.151    Comp.152
## Standard deviation     0.131247516 0.131216935 0.131163391 0.131125014
## Proportion of Variance 0.001548172 0.001547451 0.001546188 0.001545284
## Cumulative Proportion  0.350428788 0.351976239 0.353522427 0.355067711
##                           Comp.153    Comp.154    Comp.155    Comp.156
## Standard deviation     0.131108115 0.131085426 0.131047772 0.131042331
## Proportion of Variance 0.001544885 0.001544351 0.001543464 0.001543335
## Cumulative Proportion  0.356612596 0.358156947 0.359700410 0.361243746
##                           Comp.157    Comp.158    Comp.159   Comp.160
## Standard deviation     0.130936156 0.130923310 0.130849871 0.13083433
## Proportion of Variance 0.001540835 0.001540533 0.001538805 0.00153844
## Cumulative Proportion  0.362784581 0.364325114 0.365863920 0.36740236
##                           Comp.161    Comp.162    Comp.163    Comp.164
## Standard deviation     0.130827879 0.130787935 0.130727402 0.130697877
## Proportion of Variance 0.001538288 0.001537349 0.001535926 0.001535232
## Cumulative Proportion  0.368940647 0.370477996 0.372013923 0.373549155
##                           Comp.165    Comp.166    Comp.167    Comp.168
## Standard deviation     0.130668559 0.130652183 0.130562519 0.130553776
## Proportion of Variance 0.001534544 0.001534159 0.001532054 0.001531849
## Cumulative Proportion  0.375083699 0.376617858 0.378149912 0.379681761
##                           Comp.169   Comp.170    Comp.171    Comp.172
## Standard deviation     0.130543710 0.13051802 0.130474422 0.130405322
## Proportion of Variance 0.001531613 0.00153101 0.001529987 0.001528367
## Cumulative Proportion  0.381213374 0.38274438 0.384274371 0.385802739
##                          Comp.173    Comp.174    Comp.175    Comp.176
## Standard deviation     0.13038197 0.130349750 0.130320193 0.130264482
## Proportion of Variance 0.00152782 0.001527065 0.001526372 0.001525068
## Cumulative Proportion  0.38733056 0.388857623 0.390383996 0.391909063
##                          Comp.177    Comp.178    Comp.179    Comp.180
## Standard deviation     0.13024579 0.130221776 0.130205817 0.130161446
## Proportion of Variance 0.00152463 0.001524068 0.001523694 0.001522656
## Cumulative Proportion  0.39343369 0.394957761 0.396481456 0.398004112
##                           Comp.181    Comp.182    Comp.183    Comp.184
## Standard deviation     0.130132895 0.130072206 0.130061064 0.130050247
## Proportion of Variance 0.001521988 0.001520569 0.001520308 0.001520056
## Cumulative Proportion  0.399526100 0.401046669 0.402566977 0.404087033
##                           Comp.185    Comp.186    Comp.187    Comp.188
## Standard deviation     0.130015096 0.130010105 0.129992438 0.129938114
## Proportion of Variance 0.001519234 0.001519117 0.001518704 0.001517435
## Cumulative Proportion  0.405606267 0.407125384 0.408644088 0.410161524
##                           Comp.189   Comp.190   Comp.191    Comp.192
## Standard deviation     0.129913718 0.12988651 0.12987751 0.129861812
## Proportion of Variance 0.001516866 0.00151623 0.00151602 0.001515654
## Cumulative Proportion  0.411678389 0.41319462 0.41471064 0.416226294
##                           Comp.193    Comp.194    Comp.195    Comp.196
## Standard deviation     0.129787358 0.129753754 0.129722296 0.129667051
## Proportion of Variance 0.001513916 0.001513132 0.001512399 0.001511111
## Cumulative Proportion  0.417740210 0.419253342 0.420765741 0.422276852
##                           Comp.197   Comp.198    Comp.199    Comp.200
## Standard deviation     0.129648942 0.12961680 0.129553388 0.129546425
## Proportion of Variance 0.001510689 0.00150994 0.001508463 0.001508301
## Cumulative Proportion  0.423787541 0.42529748 0.426805944 0.428314245
##                           Comp.201    Comp.202    Comp.203    Comp.204
## Standard deviation     0.129528563 0.129517123 0.129470253 0.129446345
## Proportion of Variance 0.001507885 0.001507619 0.001506528 0.001505971
## Cumulative Proportion  0.429822129 0.431329748 0.432836275 0.434342247
##                          Comp.205    Comp.206    Comp.207    Comp.208
## Standard deviation     0.12940544 0.129377037 0.129337587 0.129287399
## Proportion of Variance 0.00150502 0.001504359 0.001503442 0.001502275
## Cumulative Proportion  0.43584727 0.437351625 0.438855067 0.440357342
##                          Comp.209   Comp.210    Comp.211    Comp.212
## Standard deviation     0.12925534 0.12921485 0.129167852 0.129140254
## Proportion of Variance 0.00150153 0.00150059 0.001499498 0.001498858
## Cumulative Proportion  0.44185887 0.44335946 0.444858960 0.446357818
##                           Comp.213    Comp.214    Comp.215    Comp.216
## Standard deviation     0.129119772 0.129113573 0.129074390 0.129056729
## Proportion of Variance 0.001498382 0.001498238 0.001497329 0.001496919
## Cumulative Proportion  0.447856200 0.449354438 0.450851767 0.452348687
##                           Comp.217   Comp.218    Comp.219    Comp.220
## Standard deviation     0.129014871 0.12899078 0.128945732 0.128925952
## Proportion of Variance 0.001495948 0.00149539 0.001494346 0.001493887
## Cumulative Proportion  0.453844635 0.45534002 0.456834370 0.458328258
##                           Comp.221    Comp.222    Comp.223    Comp.224
## Standard deviation     0.128878220 0.128864072 0.128833230 0.128777778
## Proportion of Variance 0.001492781 0.001492453 0.001491739 0.001490455
## Cumulative Proportion  0.459821039 0.461313492 0.462805231 0.464295686
##                           Comp.225    Comp.226    Comp.227    Comp.228
## Standard deviation     0.128757220 0.128701662 0.128680424 0.128624294
## Proportion of Variance 0.001489979 0.001488694 0.001488203 0.001486905
## Cumulative Proportion  0.465785666 0.467274360 0.468762562 0.470249467
##                           Comp.229    Comp.230    Comp.231    Comp.232
## Standard deviation     0.128608189 0.128598832 0.128587176 0.128511970
## Proportion of Variance 0.001486532 0.001486316 0.001486047 0.001484309
## Cumulative Proportion  0.471735999 0.473222315 0.474708361 0.476192670
##                           Comp.233    Comp.234    Comp.235    Comp.236
## Standard deviation     0.128501232 0.128464100 0.128429033 0.128398613
## Proportion of Variance 0.001484061 0.001483203 0.001482394 0.001481691
## Cumulative Proportion  0.477676731 0.479159934 0.480642328 0.482124019
##                           Comp.237    Comp.238    Comp.239    Comp.240
## Standard deviation     0.128376790 0.128359199 0.128293151 0.128265625
## Proportion of Variance 0.001481188 0.001480782 0.001479258 0.001478624
## Cumulative Proportion  0.483605207 0.485085989 0.486565247 0.488043871
##                           Comp.241    Comp.242   Comp.243    Comp.244
## Standard deviation     0.128226511 0.128217874 0.12820473 0.128178817
## Proportion of Variance 0.001477722 0.001477523 0.00147722 0.001476623
## Cumulative Proportion  0.489521593 0.490999115 0.49247634 0.493952958
##                           Comp.245    Comp.246    Comp.247    Comp.248
## Standard deviation     0.128158000 0.128135415 0.128085204 0.128068903
## Proportion of Variance 0.001476143 0.001475623 0.001474467 0.001474092
## Cumulative Proportion  0.495429102 0.496904725 0.498379192 0.499853283
##                           Comp.249    Comp.250    Comp.251    Comp.252
## Standard deviation     0.128037601 0.127991408 0.127958508 0.127926209
## Proportion of Variance 0.001473371 0.001472308 0.001471551 0.001470809
## Cumulative Proportion  0.501326654 0.502798962 0.504270514 0.505741322
##                           Comp.253    Comp.254    Comp.255    Comp.256
## Standard deviation     0.127861294 0.127818194 0.127776749 0.127761853
## Proportion of Variance 0.001469316 0.001468326 0.001467374 0.001467032
## Cumulative Proportion  0.507210638 0.508678964 0.510146338 0.511613370
##                           Comp.257    Comp.258    Comp.259    Comp.260
## Standard deviation     0.127742895 0.127731540 0.127670167 0.127664386
## Proportion of Variance 0.001466596 0.001466336 0.001464927 0.001464794
## Cumulative Proportion  0.513079966 0.514546302 0.516011229 0.517476023
##                           Comp.261    Comp.262    Comp.263    Comp.264
## Standard deviation     0.127615363 0.127583768 0.127539557 0.127533315
## Proportion of Variance 0.001463669 0.001462945 0.001461931 0.001461788
## Cumulative Proportion  0.518939692 0.520402637 0.521864568 0.523326356
##                          Comp.265    Comp.266    Comp.267    Comp.268
## Standard deviation     0.12747799 0.127455763 0.127413518 0.127392457
## Proportion of Variance 0.00146052 0.001460011 0.001459043 0.001458561
## Cumulative Proportion  0.52478688 0.526246887 0.527705930 0.529164490
##                          Comp.269    Comp.270    Comp.271    Comp.272
## Standard deviation     0.12736493 0.127338462 0.127317203 0.127287323
## Proportion of Variance 0.00145793 0.001457325 0.001456838 0.001456154
## Cumulative Proportion  0.53062242 0.532079745 0.533536583 0.534992737
##                           Comp.273    Comp.274    Comp.275    Comp.276
## Standard deviation     0.127216156 0.127184352 0.127158166 0.127117662
## Proportion of Variance 0.001454526 0.001453799 0.001453201 0.001452275
## Cumulative Proportion  0.536447264 0.537901063 0.539354264 0.540806539
##                           Comp.277    Comp.278    Comp.279    Comp.280
## Standard deviation     0.127077569 0.127067233 0.127016675 0.126978309
## Proportion of Variance 0.001451359 0.001451123 0.001449968 0.001449093
## Cumulative Proportion  0.542257898 0.543709021 0.545158990 0.546608082
##                           Comp.281   Comp.282    Comp.283    Comp.284
## Standard deviation     0.126970098 0.12693262 0.126922476 0.126887841
## Proportion of Variance 0.001448905 0.00144805 0.001447819 0.001447029
## Cumulative Proportion  0.548056987 0.54950504 0.550952856 0.552399885
##                           Comp.285    Comp.286    Comp.287    Comp.288
## Standard deviation     0.126864060 0.126802452 0.126779705 0.126761963
## Proportion of Variance 0.001446486 0.001445082 0.001444563 0.001444159
## Cumulative Proportion  0.553846371 0.555291453 0.556736016 0.558180175
##                           Comp.289    Comp.290    Comp.291    Comp.292
## Standard deviation     0.126724769 0.126695869 0.126688858 0.126624824
## Proportion of Variance 0.001443312 0.001442653 0.001442494 0.001441036
## Cumulative Proportion  0.559623486 0.561066140 0.562508633 0.563949669
##                          Comp.293    Comp.294    Comp.295    Comp.296
## Standard deviation     0.12659470 0.126571896 0.126526472 0.126514369
## Proportion of Variance 0.00144035 0.001439831 0.001438798 0.001438523
## Cumulative Proportion  0.56539002 0.566829851 0.568268649 0.569707172
##                           Comp.297    Comp.298    Comp.299    Comp.300
## Standard deviation     0.126468281 0.126436431 0.126385409 0.126363056
## Proportion of Variance 0.001437475 0.001436751 0.001435592 0.001435084
## Cumulative Proportion  0.571144647 0.572581398 0.574016990 0.575452074
##                           Comp.301   Comp.302    Comp.303   Comp.304
## Standard deviation     0.126315067 0.12626774 0.126250621 0.12621661
## Proportion of Variance 0.001433994 0.00143292 0.001432531 0.00143176
## Cumulative Proportion  0.576886068 0.57831899 0.579751519 0.58118328
##                           Comp.305    Comp.306    Comp.307    Comp.308
## Standard deviation     0.126173127 0.126155973 0.126151259 0.126126583
## Proportion of Variance 0.001430773 0.001430384 0.001430277 0.001429718
## Cumulative Proportion  0.582614052 0.584044436 0.585474714 0.586904432
##                           Comp.309    Comp.310    Comp.311    Comp.312
## Standard deviation     0.126108074 0.126055943 0.126040140 0.125980633
## Proportion of Variance 0.001429298 0.001428117 0.001427759 0.001426411
## Cumulative Proportion  0.588333730 0.589761847 0.591189605 0.592616016
##                           Comp.313   Comp.314    Comp.315    Comp.316
## Standard deviation     0.125936313 0.12591741 0.125885957 0.125867538
## Proportion of Variance 0.001425407 0.00142498 0.001424268 0.001423851
## Cumulative Proportion  0.594041424 0.59546640 0.596890671 0.598314522
##                           Comp.317   Comp.318    Comp.319    Comp.320
## Standard deviation     0.125844817 0.12579939 0.125788924 0.125779724
## Proportion of Variance 0.001423337 0.00142231 0.001422073 0.001421865
## Cumulative Proportion  0.599737859 0.60116017 0.602582242 0.604004107
##                          Comp.321    Comp.322    Comp.323    Comp.324
## Standard deviation     0.12573351 0.125713893 0.125694636 0.125656107
## Proportion of Variance 0.00142082 0.001420377 0.001419942 0.001419072
## Cumulative Proportion  0.60542493 0.606845304 0.608265246 0.609684318
##                           Comp.325    Comp.326    Comp.327    Comp.328
## Standard deviation     0.125573969 0.125533622 0.125474032 0.125451598
## Proportion of Variance 0.001417217 0.001416306 0.001414962 0.001414456
## Cumulative Proportion  0.611101535 0.612517841 0.613932803 0.615347259
##                           Comp.329    Comp.330    Comp.331    Comp.332
## Standard deviation     0.125425497 0.125417871 0.125380202 0.125343313
## Proportion of Variance 0.001413868 0.001413696 0.001412847 0.001412015
## Cumulative Proportion  0.616761127 0.618174822 0.619587669 0.620999684
##                           Comp.333   Comp.334    Comp.335    Comp.336
## Standard deviation     0.125338483 0.12526981 0.125247094 0.125211530
## Proportion of Variance 0.001411907 0.00141036 0.001409848 0.001409048
## Cumulative Proportion  0.622411591 0.62382195 0.625231799 0.626640847
##                           Comp.337   Comp.338    Comp.339    Comp.340
## Standard deviation     0.125197795 0.12513874 0.125118511 0.125074763
## Proportion of Variance 0.001408739 0.00140741 0.001406955 0.001405971
## Cumulative Proportion  0.628049586 0.62945700 0.630863951 0.632269922
##                           Comp.341    Comp.342    Comp.343   Comp.344
## Standard deviation     0.125031646 0.124985569 0.124973043 0.12496170
## Proportion of Variance 0.001405002 0.001403967 0.001403685 0.00140343
## Cumulative Proportion  0.633674924 0.635078891 0.636482576 0.63788601
##                           Comp.345   Comp.346    Comp.347    Comp.348
## Standard deviation     0.124946198 0.12492115 0.124874000 0.124849102
## Proportion of Variance 0.001403082 0.00140252 0.001401461 0.001400903
## Cumulative Proportion  0.639289089 0.64069161 0.642093070 0.643493973
##                           Comp.349    Comp.350    Comp.351    Comp.352
## Standard deviation     0.124829788 0.124801061 0.124773247 0.124735535
## Proportion of Variance 0.001400469 0.001399825 0.001399201 0.001398355
## Cumulative Proportion  0.644894442 0.646294267 0.647693467 0.649091823
##                           Comp.353    Comp.354    Comp.355    Comp.356
## Standard deviation     0.124725684 0.124680981 0.124656507 0.124606858
## Proportion of Variance 0.001398134 0.001397132 0.001396584 0.001395472
## Cumulative Proportion  0.650489957 0.651887089 0.653283673 0.654679144
##                           Comp.357    Comp.358    Comp.359    Comp.360
## Standard deviation     0.124559659 0.124513626 0.124502435 0.124485007
## Proportion of Variance 0.001394415 0.001393384 0.001393134 0.001392744
## Cumulative Proportion  0.656073559 0.657466943 0.658860077 0.660252820
##                           Comp.361    Comp.362    Comp.363   Comp.364
## Standard deviation     0.124453142 0.124404936 0.124398192 0.12436545
## Proportion of Variance 0.001392031 0.001390953 0.001390802 0.00139007
## Cumulative Proportion  0.661644851 0.663035803 0.664426605 0.66581667
##                           Comp.365    Comp.366   Comp.367    Comp.368
## Standard deviation     0.124324522 0.124297939 0.12427863 0.124216178
## Proportion of Variance 0.001389155 0.001388561 0.00138813 0.001386735
## Cumulative Proportion  0.667205830 0.668594391 0.66998252 0.671369255
##                           Comp.369   Comp.370    Comp.371    Comp.372
## Standard deviation     0.124176477 0.12415370 0.124128852 0.124066103
## Proportion of Variance 0.001385848 0.00138534 0.001384786 0.001383386
## Cumulative Proportion  0.672755103 0.67414044 0.675525229 0.676908615
##                           Comp.373    Comp.374    Comp.375    Comp.376
## Standard deviation     0.124038671 0.123991770 0.123980521 0.123946124
## Proportion of Variance 0.001382774 0.001381729 0.001381478 0.001380712
## Cumulative Proportion  0.678291389 0.679673118 0.681054596 0.682435308
##                           Comp.377    Comp.378    Comp.379    Comp.380
## Standard deviation     0.123923756 0.123914085 0.123881739 0.123848020
## Proportion of Variance 0.001380213 0.001379998 0.001379278 0.001378527
## Cumulative Proportion  0.683815521 0.685195519 0.686574797 0.687953324
##                           Comp.381    Comp.382    Comp.383    Comp.384
## Standard deviation     0.123826639 0.123777266 0.123766103 0.123721262
## Proportion of Variance 0.001378051 0.001376952 0.001376704 0.001375706
## Cumulative Proportion  0.689331375 0.690708327 0.692085031 0.693460737
##                           Comp.385    Comp.386    Comp.387    Comp.388
## Standard deviation     0.123657786 0.123625583 0.123609026 0.123583067
## Proportion of Variance 0.001374295 0.001373579 0.001373212 0.001372635
## Cumulative Proportion  0.694835032 0.696208612 0.697581823 0.698954458
##                         Comp.389    Comp.390    Comp.391   Comp.392
## Standard deviation     0.1235500 0.123523169 0.123511522 0.12346799
## Proportion of Variance 0.0013719 0.001371305 0.001371046 0.00137008
## Cumulative Proportion  0.7003264 0.701697662 0.703068708 0.70443879
##                           Comp.393    Comp.394    Comp.395    Comp.396
## Standard deviation     0.123444842 0.123411552 0.123349070 0.123323612
## Proportion of Variance 0.001369566 0.001368827 0.001367442 0.001366877
## Cumulative Proportion  0.705808354 0.707177182 0.708544623 0.709911501
##                          Comp.397    Comp.398    Comp.399   Comp.400
## Standard deviation     0.12329487 0.123284975 0.123231400 0.12321814
## Proportion of Variance 0.00136624 0.001366021 0.001364834 0.00136454
## Cumulative Proportion  0.71127774 0.712643762 0.714008596 0.71537314
##                           Comp.401   Comp.402    Comp.403    Comp.404
## Standard deviation     0.123199956 0.12315851 0.123129372 0.123094140
## Proportion of Variance 0.001364138 0.00136322 0.001362575 0.001361795
## Cumulative Proportion  0.716737274 0.71810049 0.719463069 0.720824864
##                           Comp.405    Comp.406    Comp.407    Comp.408
## Standard deviation     0.123081006 0.123036263 0.123016834 0.122972076
## Proportion of Variance 0.001361505 0.001360515 0.001360085 0.001359096
## Cumulative Proportion  0.722186369 0.723546884 0.724906969 0.726266065
##                           Comp.409    Comp.410    Comp.411    Comp.412
## Standard deviation     0.122895272 0.122882499 0.122859031 0.122851160
## Proportion of Variance 0.001357399 0.001357117 0.001356598 0.001356424
## Cumulative Proportion  0.727623464 0.728980581 0.730337179 0.731693603
##                           Comp.413    Comp.414    Comp.415  Comp.416
## Standard deviation     0.122811317 0.122771787 0.122726744 0.1226779
## Proportion of Variance 0.001355545 0.001354672 0.001353678 0.0013526
## Cumulative Proportion  0.733049148 0.734403820 0.735757499 0.7371101
##                           Comp.417    Comp.418    Comp.419    Comp.420
## Standard deviation     0.122644439 0.122607602 0.122566359 0.122528814
## Proportion of Variance 0.001351863 0.001351051 0.001350143 0.001349316
## Cumulative Proportion  0.738461963 0.739813014 0.741163157 0.742512472
##                           Comp.421  Comp.422    Comp.423    Comp.424
## Standard deviation     0.122470205 0.1224554 0.122405082 0.122371266
## Proportion of Variance 0.001348025 0.0013477 0.001346592 0.001345848
## Cumulative Proportion  0.743860497 0.7452082 0.746554789 0.747900637
##                           Comp.425    Comp.426    Comp.427    Comp.428
## Standard deviation     0.122338105 0.122276200 0.122232946 0.122230062
## Proportion of Variance 0.001345119 0.001343758 0.001342807 0.001342744
## Cumulative Proportion  0.749245756 0.750589513 0.751932320 0.753275064
##                           Comp.429    Comp.430    Comp.431    Comp.432
## Standard deviation     0.122178160 0.122143026 0.122084045 0.122043775
## Proportion of Variance 0.001341604 0.001340832 0.001339538 0.001338654
## Cumulative Proportion  0.754616668 0.755957500 0.757297038 0.758635692
##                           Comp.433    Comp.434    Comp.435    Comp.436
## Standard deviation     0.122029119 0.122001490 0.121980438 0.121944981
## Proportion of Variance 0.001338333 0.001337727 0.001337265 0.001336488
## Cumulative Proportion  0.759974024 0.761311751 0.762649016 0.763985503
##                          Comp.437    Comp.438    Comp.439    Comp.440
## Standard deviation     0.12188712 0.121885351 0.121864137 0.121824306
## Proportion of Variance 0.00133522 0.001335181 0.001334716 0.001333844
## Cumulative Proportion  0.76532072 0.766655904 0.767990620 0.769324464
##                           Comp.441    Comp.442    Comp.443    Comp.444
## Standard deviation     0.121755742 0.121723268 0.121684732 0.121662402
## Proportion of Variance 0.001332343 0.001331632 0.001330789 0.001330301
## Cumulative Proportion  0.770656807 0.771988439 0.773319228 0.774649529
##                           Comp.445    Comp.446    Comp.447    Comp.448
## Standard deviation     0.121629260 0.121600910 0.121567836 0.121489476
## Proportion of Variance 0.001329576 0.001328956 0.001328234 0.001326522
## Cumulative Proportion  0.775979105 0.777308061 0.778636295 0.779962817
##                          Comp.449   Comp.450  Comp.451    Comp.452
## Standard deviation     0.12147702 0.12144587 0.1214198 0.121385907
## Proportion of Variance 0.00132625 0.00132557 0.0013250 0.001324261
## Cumulative Proportion  0.78128907 0.78261464 0.7839396 0.785263897
##                           Comp.453    Comp.454    Comp.455    Comp.456
## Standard deviation     0.121378228 0.121319527 0.121279196 0.121199874
## Proportion of Variance 0.001324094 0.001322813 0.001321934 0.001320205
## Cumulative Proportion  0.786587991 0.787910804 0.789232738 0.790552943
##                           Comp.457    Comp.458    Comp.459   Comp.460
## Standard deviation     0.121154479 0.121087962 0.121080909 0.12100990
## Proportion of Variance 0.001319216 0.001317768 0.001317615 0.00131607
## Cumulative Proportion  0.791872159 0.793189927 0.794507542 0.79582361
##                           Comp.461    Comp.462   Comp.463    Comp.464
## Standard deviation     0.120993719 0.120966306 0.12091747 0.120877512
## Proportion of Variance 0.001315718 0.001315122 0.00131406 0.001313192
## Cumulative Proportion  0.797139330 0.798454451 0.79976851 0.801081703
##                          Comp.465    Comp.466    Comp.467    Comp.468
## Standard deviation     0.12082543 0.120818800 0.120757115 0.120743185
## Proportion of Variance 0.00131206 0.001311916 0.001310577 0.001310275
## Cumulative Proportion  0.80239376 0.803705679 0.805016256 0.806326531
##                           Comp.469    Comp.470    Comp.471    Comp.472
## Standard deviation     0.120704411 0.120667820 0.120640427 0.120539810
## Proportion of Variance 0.001309433 0.001308639 0.001308045 0.001305864
## Cumulative Proportion  0.807635964 0.808944604 0.810252649 0.811558513
##                           Comp.473    Comp.474    Comp.475    Comp.476
## Standard deviation     0.120518805 0.120496525 0.120465011 0.120450201
## Proportion of Variance 0.001305409 0.001304927 0.001304244 0.001303924
## Cumulative Proportion  0.812863923 0.814168850 0.815473094 0.816777017
##                           Comp.477    Comp.478   Comp.479    Comp.480
## Standard deviation     0.120401684 0.120375607 0.12034421 0.120294140
## Proportion of Variance 0.001302873 0.001302309 0.00130163 0.001300547
## Cumulative Proportion  0.818079891 0.819382200 0.82068383 0.821984377
##                           Comp.481    Comp.482    Comp.483    Comp.484
## Standard deviation     0.120259751 0.120226106 0.120186068 0.120095658
## Proportion of Variance 0.001299803 0.001299076 0.001298211 0.001296259
## Cumulative Proportion  0.823284180 0.824583256 0.825881467 0.827177726
##                          Comp.485    Comp.486    Comp.487    Comp.488
## Standard deviation     0.12007671 0.120003846 0.119998669 0.119943608
## Proportion of Variance 0.00129585 0.001294278 0.001294166 0.001292978
## Cumulative Proportion  0.82847358 0.829767853 0.831062019 0.832354998
##                           Comp.489    Comp.490    Comp.491    Comp.492
## Standard deviation     0.119889785 0.119861898 0.119845616 0.119794985
## Proportion of Variance 0.001291818 0.001291217 0.001290867 0.001289776
## Cumulative Proportion  0.833646816 0.834938034 0.836228900 0.837518676
##                           Comp.493    Comp.494    Comp.495    Comp.496
## Standard deviation     0.119744571 0.119715536 0.119707700 0.119648121
## Proportion of Variance 0.001288691 0.001288066 0.001287897 0.001286616
## Cumulative Proportion  0.838807367 0.840095433 0.841383331 0.842669946
##                           Comp.497    Comp.498    Comp.499   Comp.500
## Standard deviation     0.119635525 0.119610826 0.119555890 0.11953902
## Proportion of Variance 0.001286345 0.001285814 0.001284633 0.00128427
## Cumulative Proportion  0.843956291 0.845242105 0.846526738 0.84781101
##                           Comp.501    Comp.502    Comp.503    Comp.504
## Standard deviation     0.119524479 0.119455476 0.119425447 0.119414903
## Proportion of Variance 0.001283958 0.001282476 0.001281831 0.001281605
## Cumulative Proportion  0.849094966 0.850377442 0.851659273 0.852940878
##                           Comp.505    Comp.506   Comp.507   Comp.508
## Standard deviation     0.119357538 0.119342493 0.11931167 0.11926922
## Proportion of Variance 0.001280374 0.001280051 0.00127939 0.00127848
## Cumulative Proportion  0.854221252 0.855501303 0.85678069 0.85805917
##                           Comp.509    Comp.510   Comp.511    Comp.512
## Standard deviation     0.119261055 0.119192975 0.11915119 0.119127601
## Proportion of Variance 0.001278305 0.001276846 0.00127595 0.001275445
## Cumulative Proportion  0.859337477 0.860614323 0.86189027 0.863165719
##                           Comp.513    Comp.514    Comp.515    Comp.516
## Standard deviation     0.119080929 0.119045248 0.119003164 0.118985945
## Proportion of Variance 0.001274446 0.001273683 0.001272782 0.001272414
## Cumulative Proportion  0.864440165 0.865713848 0.866986630 0.868259044
##                           Comp.517    Comp.518    Comp.519    Comp.520
## Standard deviation     0.118936891 0.118860491 0.118827727 0.118808615
## Proportion of Variance 0.001271365 0.001269732 0.001269032 0.001268624
## Cumulative Proportion  0.869530409 0.870800141 0.872069173 0.873337797
##                           Comp.521    Comp.522    Comp.523    Comp.524
## Standard deviation     0.118729339 0.118713192 0.118697554 0.118649373
## Proportion of Variance 0.001266932 0.001266587 0.001266253 0.001265226
## Cumulative Proportion  0.874604729 0.875871316 0.877137569 0.878402795
##                           Comp.525    Comp.526    Comp.527    Comp.528
## Standard deviation     0.118605389 0.118599606 0.118516561 0.118457281
## Proportion of Variance 0.001264288 0.001264164 0.001262395 0.001261132
## Cumulative Proportion  0.879667083 0.880931247 0.882193642 0.883454774
##                           Comp.529   Comp.530    Comp.531    Comp.532
## Standard deviation     0.118432976 0.11837075 0.118354814 0.118316850
## Proportion of Variance 0.001260615 0.00125929 0.001258951 0.001258144
## Cumulative Proportion  0.884715389 0.88597468 0.887233630 0.888491774
##                           Comp.533    Comp.534    Comp.535   Comp.536
## Standard deviation     0.118291863 0.118209726 0.118205223 0.11816848
## Proportion of Variance 0.001257612 0.001255867 0.001255771 0.00125499
## Cumulative Proportion  0.889749387 0.891005253 0.892261024 0.89351601
##                          Comp.537    Comp.538    Comp.539    Comp.540
## Standard deviation     0.11811810 0.118098051 0.118058356 0.117990889
## Proportion of Variance 0.00125392 0.001253495 0.001252652 0.001251221
## Cumulative Proportion  0.89476993 0.896023430 0.897276082 0.898527303
##                           Comp.541   Comp.542    Comp.543    Comp.544
## Standard deviation     0.117975910 0.11794463 0.117854638 0.117823297
## Proportion of Variance 0.001250903 0.00125024 0.001248333 0.001247669
## Cumulative Proportion  0.899778206 0.90102845 0.902276779 0.903524449
##                           Comp.545    Comp.546    Comp.547    Comp.548
## Standard deviation     0.117763481 0.117722642 0.117603135 0.117558558
## Proportion of Variance 0.001246403 0.001245538 0.001243011 0.001242069
## Cumulative Proportion  0.904770851 0.906016390 0.907259400 0.908501469
##                           Comp.549    Comp.550    Comp.551    Comp.552
## Standard deviation     0.117522725 0.117468921 0.117411170 0.117336437
## Proportion of Variance 0.001241312 0.001240175 0.001238956 0.001237379
## Cumulative Proportion  0.909742780 0.910982956 0.912221912 0.913459291
##                          Comp.553    Comp.554    Comp.555    Comp.556
## Standard deviation     0.11730372 0.117284680 0.117200694 0.117191499
## Proportion of Variance 0.00123669 0.001236288 0.001234518 0.001234324
## Cumulative Proportion  0.91469598 0.915932269 0.917166787 0.918401111
##                           Comp.557   Comp.558    Comp.559    Comp.560
## Standard deviation     0.117171729 0.11711387 0.117071658 0.117055355
## Proportion of Variance 0.001233908 0.00123269 0.001231801 0.001231458
## Cumulative Proportion  0.919635019 0.92086771 0.922099510 0.923330968
##                           Comp.561    Comp.562    Comp.563    Comp.564
## Standard deviation     0.117002902 0.116916393 0.116873112 0.116794814
## Proportion of Variance 0.001230355 0.001228536 0.001227627 0.001225982
## Cumulative Proportion  0.924561323 0.925789859 0.927017485 0.928243468
##                           Comp.565    Comp.566    Comp.567    Comp.568
## Standard deviation     0.116770737 0.116715281 0.116644011 0.116532957
## Proportion of Variance 0.001225477 0.001224313 0.001222818 0.001220491
## Cumulative Proportion  0.929468944 0.930693258 0.931916076 0.933136567
##                           Comp.569    Comp.570    Comp.571   Comp.572
## Standard deviation     0.116531028 0.116442987 0.116400711 0.11633795
## Proportion of Variance 0.001220451 0.001218607 0.001217723 0.00121641
## Cumulative Proportion  0.934357018 0.935575625 0.936793348 0.93800976
##                           Comp.573    Comp.574  Comp.575    Comp.576
## Standard deviation     0.116323456 0.116267976 0.1162274 0.116130516
## Proportion of Variance 0.001216107 0.001214947 0.0012141 0.001212076
## Cumulative Proportion  0.939225864 0.940440811 0.9416549 0.942866987
##                           Comp.577    Comp.578    Comp.579    Comp.580
## Standard deviation     0.116085019 0.116028358 0.115979210 0.115971021
## Proportion of Variance 0.001211126 0.001209944 0.001208919 0.001208749
## Cumulative Proportion  0.944078113 0.945288058 0.946496977 0.947705726
##                           Comp.581    Comp.582    Comp.583   Comp.584
## Standard deviation     0.115885696 0.115844672 0.115795182 0.11568192
## Proportion of Variance 0.001206971 0.001206116 0.001205086 0.00120273
## Cumulative Proportion  0.948912697 0.950118813 0.951323899 0.95252663
##                           Comp.585    Comp.586    Comp.587    Comp.588
## Standard deviation     0.115649051 0.115536977 0.115461515 0.115385823
## Proportion of Variance 0.001202046 0.001199718 0.001198151 0.001196581
## Cumulative Proportion  0.953728675 0.954928393 0.956126544 0.957323125
##                           Comp.589    Comp.590    Comp.591    Comp.592
## Standard deviation     0.115236619 0.115159124 0.115156157 0.115090774
## Proportion of Variance 0.001193488 0.001191883 0.001191822 0.001190469
## Cumulative Proportion  0.958516613 0.959708496 0.960900318 0.962090787
##                           Comp.593    Comp.594    Comp.595    Comp.596
## Standard deviation     0.115072391 0.115038219 0.114943704 0.114573323
## Proportion of Variance 0.001190089 0.001189382 0.001187428 0.001179788
## Cumulative Proportion  0.963280876 0.964470258 0.965657687 0.966837475
##                           Comp.597   Comp.598    Comp.599    Comp.600
## Standard deviation     0.114555203 0.11442567 0.114317510 0.114254542
## Proportion of Variance 0.001179415 0.00117675 0.001174526 0.001173232
## Cumulative Proportion  0.968016890 0.96919364 0.970368166 0.971541398
##                           Comp.601    Comp.602    Comp.603    Comp.604
## Standard deviation     0.114137350 0.114072392 0.113923965 0.113836976
## Proportion of Variance 0.001170827 0.001169494 0.001166453 0.001164672
## Cumulative Proportion  0.972712225 0.973881719 0.975048172 0.976212845
##                           Comp.605   Comp.606    Comp.607    Comp.608
## Standard deviation     0.113807488 0.11370189 0.113550889 0.113504126
## Proportion of Variance 0.001164069 0.00116191 0.001158826 0.001157872
## Cumulative Proportion  0.977376914 0.97853882 0.979697650 0.980855521
##                           Comp.609    Comp.610    Comp.611    Comp.612
## Standard deviation     0.113397001 0.113364969 0.113274500 0.113117677
## Proportion of Variance 0.001155687 0.001155034 0.001153191 0.001150001
## Cumulative Proportion  0.982011208 0.983166242 0.984319434 0.985469434
##                           Comp.613    Comp.614    Comp.615    Comp.616
## Standard deviation     0.113038510 0.112628548 0.112454325 0.112406299
## Proportion of Variance 0.001148391 0.001140077 0.001136552 0.001135582
## Cumulative Proportion  0.986617826 0.987757902 0.988894455 0.990030036
##                           Comp.617    Comp.618    Comp.619    Comp.620
## Standard deviation     0.112321287 0.111584956 0.111484973 0.111203715
## Proportion of Variance 0.001133865 0.001119047 0.001117043 0.001111414
## Cumulative Proportion  0.991163901 0.992282948 0.993399991 0.994511404
##                           Comp.621    Comp.622    Comp.623    Comp.624
## Standard deviation     0.110762702 0.110688588 0.110409204 0.110390952
## Proportion of Variance 0.001102616 0.001101141 0.001095589 0.001095227
## Cumulative Proportion  0.995614020 0.996715161 0.997810749 0.998905976
##                           Comp.625
## Standard deviation     0.110330310
## Proportion of Variance 0.001094024
## Cumulative Proportion  1.000000000

The above histogram and table, show components contribution to the variance respectively. None of the components are significant as alone to the variance.

TASK 3 b Lets decide to use the first component to construct the image

As the components converts pixel values in the original image into the scores of PCA, it is possible to reconstruct the image from the scores. We expect to see higher variance explaining components will reconstruct the image closer to the unprocessed noisy image.

To reconstruct the images, we need to process scores. Scores are firstly normalized, then, turned into matrices. At last, they are displayed by the rasterImage function.

score_1comp <- image_plane_noise_gray_PCA$scores[,1]
score_2comp <- image_plane_noise_gray_PCA$scores[,2]
score_3comp <- image_plane_noise_gray_PCA$scores[,3]

score_1comp <- (score_1comp - min(score_1comp)) / (max(score_1comp)-min(score_1comp))
score_2comp <- (score_2comp - min(score_2comp))/(max(score_2comp)-min(score_2comp))
score_3comp <- (score_3comp - min(score_3comp))/(max(score_3comp)-min(score_3comp))

matrix_score_1comp <- matrix(score_1comp, 232, byrow=TRUE)
matrix_score_2comp <- matrix(score_2comp, 232, byrow=TRUE)
matrix_score_3comp <- matrix(score_3comp, 232, byrow=TRUE)

TASK 3 b plot the scores(ie mapping for the first, second and third component as an image and unprocessed)

par(mfrow = c(1,4))

plot(1:232,1:232, ann = TRUE, axes = FALSE, col = 0, main = "Unprocessed")
rasterImage(image_plane_noise_gray,0,0,232,232)
plot(x,y, ann = TRUE, axes = FALSE, col = 0, main = "Reconst. Comp 1")
rasterImage(matrix_score_1comp,0,0,232,232)
plot(x,y, ann = TRUE, axes = FALSE, col = 0, main = " Reconst. Comp 2")
rasterImage(matrix_score_2comp,0,0,232,232)
plot(x,y, ann = TRUE, axes = FALSE, col = 0, main = " Reconst. Comp3")
rasterImage(matrix_score_3comp,0,0,232,232)

A it can be seen above, the reconstruction of component 1 is slightly close to the unprocessed image compared to the others which are only the silhouettes. Please look the barplot of the variences.

TASK 3 c Plot the first, second and third component as 25 by 25 image

Eigenvector Analysis

For the transformation, the eigenvectors give inportant information, Eigenvectors are plotted how they affect the patches.

For this, the eigenvectors are normalized and convert into matrices. Finally they are displayed.

eigen_comp1 <- image_plane_noise_gray_PCA$loadings[,1]
eigen_comp2 <- image_plane_noise_gray_PCA$loadings[,2]
eigen_comp3 <- image_plane_noise_gray_PCA$loadings[,3]
eigen_comp1 <- (eigen_comp1 - min(eigen_comp1))/(max(eigen_comp1)-min(eigen_comp1))
eigen_comp2 <- (eigen_comp2 - min(eigen_comp2))/(max(eigen_comp2)-min(eigen_comp2))
eigen_comp3 <- (eigen_comp3 - min(eigen_comp3))/(max(eigen_comp3)-min(eigen_comp3))
eigen_matrix_1 <- matrix(eigen_comp1,25)
eigen_matrix_2 <- matrix(eigen_comp2,25)
eigen_matrix_3 <- matrix(eigen_comp3,25)

image_plane_noise_gray_PCA$loadings[,1:3]
##          Comp.1        Comp.2        Comp.3
## V1   0.02904268  4.261770e-02  2.840594e-02
## V2   0.02972987  4.509996e-02  2.696481e-02
## V3   0.03037982  4.727019e-02  2.452308e-02
## V4   0.03093218  4.970368e-02  2.162254e-02
## V5   0.03141521  5.127737e-02  1.719884e-02
## V6   0.03188993  5.283159e-02  1.272527e-02
## V7   0.03234619  5.373570e-02  7.818289e-03
## V8   0.03283499  5.448125e-02  1.862455e-03
## V9   0.03319305  5.458318e-02 -4.110571e-03
## V10  0.03355509  5.442919e-02 -1.060651e-02
## V11  0.03377960  5.418327e-02 -1.684265e-02
## V12  0.03402375  5.335714e-02 -2.296624e-02
## V13  0.03418382  5.238499e-02 -2.867235e-02
## V14  0.03428597  5.138797e-02 -3.369228e-02
## V15  0.03442923  4.990604e-02 -3.849030e-02
## V16  0.03444169  4.827799e-02 -4.274678e-02
## V17  0.03437252  4.600880e-02 -4.671085e-02
## V18  0.03421527  4.390947e-02 -4.936279e-02
## V19  0.03402202  4.169840e-02 -5.106352e-02
## V20  0.03396032  3.928900e-02 -5.247233e-02
## V21  0.03380890  3.670780e-02 -5.296826e-02
## V22  0.03340414  3.420523e-02 -5.272338e-02
## V23  0.03302985  3.164124e-02 -5.192229e-02
## V24  0.03269566  2.905177e-02 -4.991837e-02
## V25  0.03240015  2.656547e-02 -4.712743e-02
## V26  0.03003560  4.531602e-02  3.074685e-02
## V27  0.03079488  4.800644e-02  2.953268e-02
## V28  0.03154393  5.037482e-02  2.718963e-02
## V29  0.03217507  5.279444e-02  2.430828e-02
## V30  0.03271945  5.445232e-02  1.975169e-02
## V31  0.03325535  5.598494e-02  1.507042e-02
## V32  0.03375669  5.685819e-02  9.885728e-03
## V33  0.03428270  5.749192e-02  3.535671e-03
## V34  0.03465868  5.743306e-02 -2.962034e-03
## V35  0.03505406  5.709277e-02 -9.917190e-03
## V36  0.03527980  5.657833e-02 -1.679763e-02
## V37  0.03551949  5.546397e-02 -2.351190e-02
## V38  0.03568736  5.420270e-02 -2.994380e-02
## V39  0.03578727  5.285899e-02 -3.557689e-02
## V40  0.03591992  5.092589e-02 -4.098731e-02
## V41  0.03589425  4.880856e-02 -4.582285e-02
## V42  0.03577952  4.611398e-02 -5.025682e-02
## V43  0.03559238  4.363170e-02 -5.332387e-02
## V44  0.03535235  4.099128e-02 -5.530127e-02
## V45  0.03523839  3.823989e-02 -5.700424e-02
## V46  0.03502592  3.540775e-02 -5.769581e-02
## V47  0.03454480  3.260175e-02 -5.742523e-02
## V48  0.03411307  2.980906e-02 -5.647380e-02
## V49  0.03368591  2.700333e-02 -5.429243e-02
## V50  0.03332536  2.431421e-02 -5.116579e-02
## V51  0.03114221  4.766269e-02  3.334291e-02
## V52  0.03195778  5.051938e-02  3.225670e-02
## V53  0.03276688  5.297061e-02  3.000901e-02
## V54  0.03345047  5.538339e-02  2.714578e-02
## V55  0.03406616  5.703245e-02  2.244726e-02
## V56  0.03465236  5.850818e-02  1.749601e-02
## V57  0.03520914  5.926468e-02  1.203504e-02
## V58  0.03577270  5.974371e-02  5.365270e-03
## V59  0.03618500  5.944983e-02 -1.629988e-03
## V60  0.03659505  5.883698e-02 -9.081413e-03
## V61  0.03684669  5.794482e-02 -1.649486e-02
## V62  0.03711131  5.642758e-02 -2.377553e-02
## V63  0.03728106  5.477733e-02 -3.064511e-02
## V64  0.03736577  5.296052e-02 -3.682830e-02
## V65  0.03748771  5.053994e-02 -4.283138e-02
## V66  0.03743640  4.799654e-02 -4.808779e-02
## V67  0.03730492  4.489717e-02 -5.279511e-02
## V68  0.03706361  4.206726e-02 -5.618946e-02
## V69  0.03677677  3.909971e-02 -5.845676e-02
## V70  0.03659947  3.598404e-02 -6.024339e-02
## V71  0.03632191  3.285940e-02 -6.105743e-02
## V72  0.03578304  2.976607e-02 -6.079028e-02
## V73  0.03527232  2.685210e-02 -5.966405e-02
## V74  0.03477904  2.389157e-02 -5.725493e-02
## V75  0.03434635  2.104994e-02 -5.398913e-02
## V76  0.03201407  4.942587e-02  3.574264e-02
## V77  0.03288050  5.236654e-02  3.476569e-02
## V78  0.03374195  5.481617e-02  3.260254e-02
## V79  0.03450004  5.714613e-02  2.980086e-02
## V80  0.03519237  5.872253e-02  2.494183e-02
## V81  0.03582571  6.004734e-02  1.983218e-02
## V82  0.03640929  6.062484e-02  1.416798e-02
## V83  0.03702583  6.091939e-02  7.249237e-03
## V84  0.03745877  6.036500e-02 -9.166216e-05
## V85  0.03788511  5.942175e-02 -7.823851e-03
## V86  0.03816035  5.818268e-02 -1.571538e-02
## V87  0.03842013  5.624476e-02 -2.357078e-02
## V88  0.03859564  5.420279e-02 -3.078507e-02
## V89  0.03867349  5.194747e-02 -3.744998e-02
## V90  0.03874637  4.903305e-02 -4.392980e-02
## V91  0.03868000  4.618064e-02 -4.957271e-02
## V92  0.03850322  4.265374e-02 -5.462706e-02
## V93  0.03821950  3.945329e-02 -5.832976e-02
## V94  0.03788777  3.621367e-02 -6.083850e-02
## V95  0.03766886  3.279908e-02 -6.277666e-02
## V96  0.03735137  2.940379e-02 -6.367160e-02
## V97  0.03675658  2.613924e-02 -6.342276e-02
## V98  0.03620082  2.317123e-02 -6.217851e-02
## V99  0.03564256  2.013834e-02 -5.958278e-02
## V100 0.03513097  1.725659e-02 -5.614165e-02
## V101 0.03275141  5.065252e-02  3.768953e-02
## V102 0.03369214  5.361102e-02  3.685589e-02
## V103 0.03461128  5.596358e-02  3.482915e-02
## V104 0.03542463  5.815517e-02  3.209368e-02
## V105 0.03616382  5.956938e-02  2.730067e-02
## V106 0.03685234  6.070312e-02  2.215753e-02
## V107 0.03745384  6.100117e-02  1.638355e-02
## V108 0.03809702  6.098987e-02  9.311855e-03
## V109 0.03855827  6.011333e-02  1.795904e-03
## V110 0.03900135  5.878286e-02 -6.211694e-03
## V111 0.03928514  5.721266e-02 -1.447030e-02
## V112 0.03955250  5.487772e-02 -2.264907e-02
## V113 0.03973236  5.245248e-02 -3.024242e-02
## V114 0.03978430  4.979924e-02 -3.741854e-02
## V115 0.03982647  4.646171e-02 -4.432255e-02
## V116 0.03972782  4.324710e-02 -5.031953e-02
## V117 0.03950962  3.937021e-02 -5.561176e-02
## V118 0.03917684  3.581739e-02 -5.949301e-02
## V119 0.03881011  3.236333e-02 -6.225379e-02
## V120 0.03854538  2.871965e-02 -6.429313e-02
## V121 0.03817017  2.514816e-02 -6.534296e-02
## V122 0.03754423  2.178996e-02 -6.517469e-02
## V123 0.03693982  1.880699e-02 -6.384488e-02
## V124 0.03634199  1.584249e-02 -6.120544e-02
## V125 0.03576965  1.301952e-02 -5.772507e-02
## V126 0.03363411  5.135747e-02  3.959296e-02
## V127 0.03461455  5.419373e-02  3.891034e-02
## V128 0.03558450  5.635577e-02  3.691391e-02
## V129 0.03643178  5.831256e-02  3.416595e-02
## V130 0.03722493  5.945395e-02  2.940857e-02
## V131 0.03794409  6.025526e-02  2.425310e-02
## V132 0.03856584  6.028098e-02  1.843678e-02
## V133 0.03923665  5.989442e-02  1.133897e-02
## V134 0.03969787  5.861519e-02  3.687704e-03
## V135 0.04014457  5.685230e-02 -4.482190e-03
## V136 0.04044994  5.487760e-02 -1.287720e-02
## V137 0.04068289  5.209829e-02 -2.130458e-02
## V138 0.04085810  4.929955e-02 -2.915971e-02
## V139 0.04090064  4.619883e-02 -3.659538e-02
## V140 0.04093806  4.246362e-02 -4.378227e-02
## V141 0.04080241  3.887907e-02 -4.994658e-02
## V142 0.04056588  3.475808e-02 -5.549705e-02
## V143 0.04019693  3.094763e-02 -5.956355e-02
## V144 0.03980321  2.731056e-02 -6.245647e-02
## V145 0.03947583  2.357699e-02 -6.462529e-02
## V146 0.03905808  1.985138e-02 -6.574289e-02
## V147 0.03839723  1.645623e-02 -6.558181e-02
## V148 0.03775327  1.344439e-02 -6.418882e-02
## V149 0.03710145  1.057097e-02 -6.156884e-02
## V150 0.03648708  7.836573e-03 -5.809414e-02
## V151 0.03458771  5.154825e-02  4.114079e-02
## V152 0.03561673  5.410966e-02  4.065481e-02
## V153 0.03660054  5.598562e-02  3.887264e-02
## V154 0.03748752  5.762130e-02  3.625652e-02
## V155 0.03830416  5.839394e-02  3.164827e-02
## V156 0.03905353  5.885697e-02  2.655408e-02
## V157 0.03969007  5.848793e-02  2.074724e-02
## V158 0.04036325  5.760158e-02  1.361225e-02
## V159 0.04083559  5.592228e-02  5.971978e-03
## V160 0.04129358  5.373640e-02 -2.208180e-03
## V161 0.04159232  5.133619e-02 -1.073777e-02
## V162 0.04182625  4.812569e-02 -1.934734e-02
## V163 0.04200183  4.500196e-02 -2.733596e-02
## V164 0.04203480  4.160818e-02 -3.505448e-02
## V165 0.04205819  3.747179e-02 -4.246575e-02
## V166 0.04188991  3.359007e-02 -4.883076e-02
## V167 0.04160749  2.911096e-02 -5.452957e-02
## V168 0.04122098  2.509182e-02 -5.879452e-02
## V169 0.04078470  2.134933e-02 -6.186941e-02
## V170 0.04041502  1.754431e-02 -6.406442e-02
## V171 0.03994917  1.381101e-02 -6.521389e-02
## V172 0.03924229  1.047953e-02 -6.514688e-02
## V173 0.03856184  7.539962e-03 -6.378722e-02
## V174 0.03786756  4.884826e-03 -6.115015e-02
## V175 0.03721520  2.339973e-03 -5.764162e-02
## V176 0.03527496  5.047470e-02  4.243178e-02
## V177 0.03631464  5.266692e-02  4.214378e-02
## V178 0.03729635  5.420547e-02  4.056222e-02
## V179 0.03821339  5.546396e-02  3.810368e-02
## V180 0.03904325  5.587752e-02  3.370263e-02
## V181 0.03980285  5.589251e-02  2.879132e-02
## V182 0.04045142  5.514529e-02  2.305850e-02
## V183 0.04116456  5.382970e-02  1.624279e-02
## V184 0.04164205  5.172534e-02  8.806597e-03
## V185 0.04209944  4.917891e-02  6.261953e-04
## V186 0.04240039  4.635523e-02 -7.805390e-03
## V187 0.04262939  4.280892e-02 -1.653794e-02
## V188 0.04278376  3.934681e-02 -2.461540e-02
## V189 0.04280581  3.567150e-02 -3.236554e-02
## V190 0.04279237  3.128333e-02 -3.993835e-02
## V191 0.04262796  2.719810e-02 -4.643146e-02
## V192 0.04232091  2.261865e-02 -5.237694e-02
## V193 0.04190836  1.848936e-02 -5.677343e-02
## V194 0.04143288  1.467624e-02 -6.005813e-02
## V195 0.04102071  1.086568e-02 -6.248397e-02
## V196 0.04050814  7.102648e-03 -6.366844e-02
## V197 0.03980314  3.832563e-03 -6.379593e-02
## V198 0.03906566  1.029201e-03 -6.254348e-02
## V199 0.03833003 -1.435432e-03 -6.000059e-02
## V200 0.03764434 -3.788861e-03 -5.661807e-02
## V201 0.03578247  4.860865e-02  4.372302e-02
## V202 0.03683349  5.045619e-02  4.372126e-02
## V203 0.03782327  5.157599e-02  4.243931e-02
## V204 0.03875649  5.235435e-02  4.030308e-02
## V205 0.03958302  5.229636e-02  3.623481e-02
## V206 0.04036127  5.191998e-02  3.156895e-02
## V207 0.04102054  5.075377e-02  2.605198e-02
## V208 0.04172779  4.893683e-02  1.947427e-02
## V209 0.04221857  4.647042e-02  1.228576e-02
## V210 0.04268801  4.358572e-02  4.179266e-03
## V211 0.04298663  4.037541e-02 -4.131910e-03
## V212 0.04319274  3.642441e-02 -1.291079e-02
## V213 0.04333066  3.263826e-02 -2.099737e-02
## V214 0.04332801  2.869087e-02 -2.879401e-02
## V215 0.04330223  2.408524e-02 -3.652169e-02
## V216 0.04313804  1.986107e-02 -4.316487e-02
## V217 0.04283044  1.517737e-02 -4.921372e-02
## V218 0.04238671  1.101152e-02 -5.371377e-02
## V219 0.04188861  7.221606e-03 -5.718851e-02
## V220 0.04144318  3.525418e-03 -5.985933e-02
## V221 0.04089390 -1.287186e-04 -6.118757e-02
## V222 0.04017534 -3.252466e-03 -6.155755e-02
## V223 0.03940774 -5.821478e-03 -6.046633e-02
## V224 0.03864760 -7.877389e-03 -5.815429e-02
## V225 0.03792157 -9.853724e-03 -5.490842e-02
## V226 0.03642223  4.612245e-02  4.478381e-02
## V227 0.03744864  4.754932e-02  4.510334e-02
## V228 0.03843935  4.829663e-02  4.410981e-02
## V229 0.03938309  4.864459e-02  4.242025e-02
## V230 0.04021300  4.819398e-02  3.871499e-02
## V231 0.04099802  4.744566e-02  3.440365e-02
## V232 0.04166029  4.592140e-02  2.919295e-02
## V233 0.04237079  4.374076e-02  2.293140e-02
## V234 0.04284653  4.087939e-02  1.591059e-02
## V235 0.04330537  3.766755e-02  7.955567e-03
## V236 0.04361220  3.408511e-02 -2.098101e-04
## V237 0.04380580  2.980396e-02 -9.017772e-03
## V238 0.04394008  2.582673e-02 -1.703648e-02
## V239 0.04392416  2.166333e-02 -2.494083e-02
## V240 0.04388503  1.691466e-02 -3.281238e-02
## V241 0.04370886  1.260220e-02 -3.968365e-02
## V242 0.04335616  7.861043e-03 -4.592066e-02
## V243 0.04291153  3.660039e-03 -5.056052e-02
## V244 0.04239740 -6.268401e-05 -5.427872e-02
## V245 0.04193150 -3.679205e-03 -5.707957e-02
## V246 0.04135905 -7.173772e-03 -5.870166e-02
## V247 0.04062840 -1.004405e-02 -5.930598e-02
## V248 0.03983793 -1.243443e-02 -5.832045e-02
## V249 0.03902103 -1.422836e-02 -5.622780e-02
## V250 0.03826710 -1.592275e-02 -5.311955e-02
## V251 0.03690576  4.247216e-02  4.609482e-02
## V252 0.03791760  4.336639e-02  4.672500e-02
## V253 0.03890753  4.372033e-02  4.614171e-02
## V254 0.03984222  4.363857e-02  4.484653e-02
## V255 0.04066081  4.270444e-02  4.154473e-02
## V256 0.04145801  4.154022e-02  3.758670e-02
## V257 0.04211577  3.968636e-02  3.272219e-02
## V258 0.04283894  3.709336e-02  2.690941e-02
## V259 0.04331497  3.396946e-02  2.028617e-02
## V260 0.04377521  3.046958e-02  1.258184e-02
## V261 0.04408854  2.663521e-02  4.620328e-03
## V262 0.04426233  2.211039e-02 -4.104872e-03
## V263 0.04440975  1.797498e-02 -1.206055e-02
## V264 0.04442014  1.362378e-02 -1.999049e-02
## V265 0.04437714  8.798345e-03 -2.785194e-02
## V266 0.04419237  4.431965e-03 -3.477320e-02
## V267 0.04381096 -2.843484e-04 -4.133306e-02
## V268 0.04334752 -4.447473e-03 -4.613323e-02
## V269 0.04282468 -8.072907e-03 -5.008102e-02
## V270 0.04235050 -1.156774e-02 -5.308718e-02
## V271 0.04175414 -1.486762e-02 -5.496122e-02
## V272 0.04100318 -1.744025e-02 -5.591042e-02
## V273 0.04018415 -1.954598e-02 -5.524863e-02
## V274 0.03936171 -2.103059e-02 -5.352966e-02
## V275 0.03856384 -2.233118e-02 -5.076500e-02
## V276 0.03743114  3.837976e-02  4.744920e-02
## V277 0.03845265  3.884891e-02  4.837734e-02
## V278 0.03944805  3.882143e-02  4.819177e-02
## V279 0.04036798  3.828861e-02  4.731666e-02
## V280 0.04119444  3.697569e-02  4.440155e-02
## V281 0.04200831  3.539617e-02  4.080008e-02
## V282 0.04265392  3.318939e-02  3.640578e-02
## V283 0.04335979  3.023657e-02  3.095678e-02
## V284 0.04383152  2.685821e-02  2.460542e-02
## V285 0.04425983  2.308921e-02  1.711739e-02
## V286 0.04455692  1.897974e-02  9.322380e-03
## V287 0.04473062  1.427360e-02  6.648213e-04
## V288 0.04488169  9.958672e-03 -7.249907e-03
## V289 0.04489337  5.498625e-03 -1.517275e-02
## V290 0.04483320  6.302068e-04 -2.304520e-02
## V291 0.04463072 -3.763055e-03 -2.999549e-02
## V292 0.04423184 -8.352324e-03 -3.675824e-02
## V293 0.04374806 -1.236188e-02 -4.182097e-02
## V294 0.04319495 -1.585143e-02 -4.607730e-02
## V295 0.04269049 -1.910779e-02 -4.927372e-02
## V296 0.04206292 -2.219046e-02 -5.148875e-02
## V297 0.04129673 -2.448729e-02 -5.281906e-02
## V298 0.04045131 -2.628151e-02 -5.244065e-02
## V299 0.03959653 -2.742441e-02 -5.110271e-02
## V300 0.03878188 -2.831179e-02 -4.859243e-02
## V301 0.03777974  3.381325e-02  4.845506e-02
## V302 0.03879742  3.383545e-02  4.974150e-02
## V303 0.03977133  3.344237e-02  4.991371e-02
## V304 0.04067112  3.249453e-02  4.946092e-02
## V305 0.04148300  3.078657e-02  4.692348e-02
## V306 0.04230339  2.884960e-02  4.378414e-02
## V307 0.04293792  2.630320e-02  3.981500e-02
## V308 0.04363072  2.308257e-02  3.473723e-02
## V309 0.04410064  1.945187e-02  2.869493e-02
## V310 0.04453717  1.546263e-02  2.157182e-02
## V311 0.04480922  1.116806e-02  1.408132e-02
## V312 0.04497594  6.285353e-03  5.651700e-03
## V313 0.04512192  1.917513e-03 -2.217961e-03
## V314 0.04511296 -2.563516e-03 -1.006348e-02
## V315 0.04504348 -7.392140e-03 -1.795099e-02
## V316 0.04484427 -1.171146e-02 -2.490728e-02
## V317 0.04443217 -1.613660e-02 -3.177682e-02
## V318 0.04394714 -1.994481e-02 -3.708596e-02
## V319 0.04340036 -2.325172e-02 -4.154342e-02
## V320 0.04287550 -2.620343e-02 -4.508403e-02
## V321 0.04223660 -2.905338e-02 -4.765463e-02
## V322 0.04147567 -3.108780e-02 -4.930190e-02
## V323 0.04060176 -3.257660e-02 -4.934282e-02
## V324 0.03974389 -3.326201e-02 -4.838941e-02
## V325 0.03890150 -3.378157e-02 -4.620285e-02
## V326 0.03804935  2.870783e-02  4.990694e-02
## V327 0.03906216  2.825725e-02  5.146425e-02
## V328 0.03999362  2.740403e-02  5.201871e-02
## V329 0.04088927  2.608500e-02  5.197543e-02
## V330 0.04169939  2.409719e-02  4.979776e-02
## V331 0.04252190  2.191757e-02  4.704345e-02
## V332 0.04314305  1.904431e-02  4.346448e-02
## V333 0.04381490  1.567139e-02  3.872098e-02
## V334 0.04426575  1.184266e-02  3.297692e-02
## V335 0.04467389  7.699615e-03  2.613043e-02
## V336 0.04495941  3.290708e-03  1.885953e-02
## V337 0.04511167 -1.662332e-03  1.063296e-02
## V338 0.04523078 -5.996128e-03  2.818969e-03
## V339 0.04521333 -1.048517e-02 -4.883972e-03
## V340 0.04513868 -1.523772e-02 -1.279329e-02
## V341 0.04493206 -1.940063e-02 -1.982703e-02
## V342 0.04450653 -2.367700e-02 -2.690773e-02
## V343 0.04402609 -2.716905e-02 -3.242372e-02
## V344 0.04346333 -3.027052e-02 -3.714683e-02
## V345 0.04290720 -3.289906e-02 -4.094554e-02
## V346 0.04227792 -3.538089e-02 -4.393417e-02
## V347 0.04150293 -3.706773e-02 -4.585608e-02
## V348 0.04062283 -3.817494e-02 -4.633570e-02
## V349 0.03976040 -3.849907e-02 -4.578271e-02
## V350 0.03887960 -3.856153e-02 -4.395754e-02
## V351 0.03830943  2.300751e-02  5.141497e-02
## V352 0.03930029  2.215631e-02  5.319334e-02
## V353 0.04020162  2.096290e-02  5.405793e-02
## V354 0.04109013  1.928027e-02  5.433102e-02
## V355 0.04189474  1.701439e-02  5.251679e-02
## V356 0.04270416  1.460100e-02  5.014153e-02
## V357 0.04331612  1.149732e-02  4.682628e-02
## V358 0.04395348  7.955725e-03  4.247705e-02
## V359 0.04438390  4.051748e-03  3.713256e-02
## V360 0.04478289 -2.450532e-04  3.061186e-02
## V361 0.04506366 -4.627139e-03  2.359666e-02
## V362 0.04518516 -9.552282e-03  1.555582e-02
## V363 0.04529126 -1.387499e-02  7.902678e-03
## V364 0.04528093 -1.824916e-02  2.388561e-04
## V365 0.04518902 -2.282588e-02 -7.600464e-03
## V366 0.04496745 -2.681901e-02 -1.459381e-02
## V367 0.04454517 -3.079921e-02 -2.178348e-02
## V368 0.04406135 -3.408027e-02 -2.749209e-02
## V369 0.04349983 -3.691674e-02 -3.248027e-02
## V370 0.04294462 -3.927131e-02 -3.671682e-02
## V371 0.04230148 -4.135478e-02 -4.005585e-02
## V372 0.04151499 -4.270985e-02 -4.235391e-02
## V373 0.04062433 -4.341718e-02 -4.323175e-02
## V374 0.03976251 -4.334557e-02 -4.315519e-02
## V375 0.03885038 -4.297008e-02 -4.173626e-02
## V376 0.03848659  1.716830e-02  5.290019e-02
## V377 0.03947667  1.589595e-02  5.495484e-02
## V378 0.04036921  1.438721e-02  5.617328e-02
## V379 0.04122711  1.236515e-02  5.680582e-02
## V380 0.04202016  9.820878e-03  5.534337e-02
## V381 0.04281802  7.218494e-03  5.326662e-02
## V382 0.04342123  3.959472e-03  5.018898e-02
## V383 0.04401926  3.144759e-04  4.610386e-02
## V384 0.04446253 -3.576259e-03  4.111498e-02
## V385 0.04485360 -7.925346e-03  3.495054e-02
## V386 0.04511152 -1.228867e-02  2.817110e-02
## V387 0.04521863 -1.710149e-02  2.030300e-02
## V388 0.04533078 -2.130531e-02  1.275301e-02
## V389 0.04532236 -2.557846e-02  5.151330e-03
## V390 0.04523245 -2.996709e-02 -2.626835e-03
## V391 0.04499171 -3.376344e-02 -9.665108e-03
## V392 0.04456290 -3.743536e-02 -1.699756e-02
## V393 0.04405340 -4.047171e-02 -2.296249e-02
## V394 0.04348948 -4.298732e-02 -2.831387e-02
## V395 0.04290624 -4.504632e-02 -3.285729e-02
## V396 0.04226516 -4.676603e-02 -3.656797e-02
## V397 0.04146486 -4.771295e-02 -3.931040e-02
## V398 0.04058016 -4.805660e-02 -4.057558e-02
## V399 0.03969878 -4.764602e-02 -4.085443e-02
## V400 0.03878259 -4.682300e-02 -3.982865e-02
## V401 0.03846109  1.126506e-02  5.410800e-02
## V402 0.03941978  9.616889e-03  5.635186e-02
## V403 0.04028624  7.911576e-03  5.779958e-02
## V404 0.04111457  5.671299e-03  5.855155e-02
## V405 0.04191201  2.902553e-03  5.736858e-02
## V406 0.04269078  1.226439e-04  5.567914e-02
## V407 0.04327548 -3.259758e-03  5.291681e-02
## V408 0.04386030 -6.911249e-03  4.915462e-02
## V409 0.04427992 -1.078875e-02  4.438987e-02
## V410 0.04465038 -1.514121e-02  3.848273e-02
## V411 0.04488862 -1.941821e-02  3.199195e-02
## V412 0.04496404 -2.407240e-02  2.435684e-02
## V413 0.04506959 -2.806366e-02  1.693869e-02
## V414 0.04502853 -3.219435e-02  9.441591e-03
## V415 0.04489968 -3.629589e-02  1.733357e-03
## V416 0.04465473 -3.978249e-02 -5.352135e-03
## V417 0.04420203 -4.318437e-02 -1.267971e-02
## V418 0.04367602 -4.584902e-02 -1.885375e-02
## V419 0.04312141 -4.806749e-02 -2.442468e-02
## V420 0.04254662 -4.980761e-02 -2.912843e-02
## V421 0.04189123 -5.108664e-02 -3.319058e-02
## V422 0.04109508 -5.167772e-02 -3.620908e-02
## V423 0.04023036 -5.163189e-02 -3.780708e-02
## V424 0.03935534 -5.090943e-02 -3.838514e-02
## V425 0.03840999 -4.970826e-02 -3.774316e-02
## V426 0.03834884  5.425719e-03  5.509543e-02
## V427 0.03929207  3.411076e-03  5.753096e-02
## V428 0.04014708  1.358253e-03  5.917800e-02
## V429 0.04094438 -1.046968e-03  6.015231e-02
## V430 0.04172105 -3.914003e-03  5.927510e-02
## V431 0.04247692 -6.814757e-03  5.779165e-02
## V432 0.04303503 -1.034970e-02  5.533417e-02
## V433 0.04359904 -1.404837e-02  5.181043e-02
## V434 0.04399557 -1.786579e-02  4.718552e-02
## V435 0.04435689 -2.213092e-02  4.151046e-02
## V436 0.04458690 -2.629130e-02  3.535708e-02
## V437 0.04463835 -3.070291e-02  2.798725e-02
## V438 0.04473501 -3.445130e-02  2.068711e-02
## V439 0.04466970 -3.835485e-02  1.331589e-02
## V440 0.04451007 -4.208042e-02  5.753235e-03
## V441 0.04424138 -4.534637e-02 -1.404189e-03
## V442 0.04380857 -4.838020e-02 -8.761693e-03
## V443 0.04325475 -5.069492e-02 -1.516863e-02
## V444 0.04269685 -5.256543e-02 -2.088044e-02
## V445 0.04211278 -5.390959e-02 -2.578433e-02
## V446 0.04145372 -5.473161e-02 -3.009445e-02
## V447 0.04065268 -5.495147e-02 -3.335967e-02
## V448 0.03978032 -5.451480e-02 -3.534625e-02
## V449 0.03891721 -5.345784e-02 -3.616288e-02
## V450 0.03796906 -5.191031e-02 -3.584240e-02
## V451 0.03826961 -3.421543e-04  5.575392e-02
## V452 0.03918155 -2.599241e-03  5.819377e-02
## V453 0.04000294 -4.837372e-03  5.991829e-02
## V454 0.04075896 -7.418525e-03  6.091595e-02
## V455 0.04153575 -1.038256e-02  6.030938e-02
## V456 0.04225334 -1.337371e-02  5.891417e-02
## V457 0.04276822 -1.692731e-02  5.667009e-02
## V458 0.04328153 -2.061389e-02  5.335971e-02
## V459 0.04364852 -2.429467e-02  4.904635e-02
## V460 0.04398016 -2.846470e-02  4.362828e-02
## V461 0.04418430 -3.245544e-02  3.773247e-02
## V462 0.04419113 -3.664834e-02  3.065919e-02
## V463 0.04425896 -4.013071e-02  2.351006e-02
## V464 0.04416260 -4.377375e-02  1.631721e-02
## V465 0.04397742 -4.715733e-02  8.957813e-03
## V466 0.04370111 -4.998973e-02  1.831486e-03
## V467 0.04326217 -5.263123e-02 -5.401177e-03
## V468 0.04270705 -5.450481e-02 -1.192708e-02
## V469 0.04217308 -5.598049e-02 -1.775382e-02
## V470 0.04157546 -5.695693e-02 -2.264387e-02
## V471 0.04091401 -5.732408e-02 -2.713229e-02
## V472 0.04011869 -5.721164e-02 -3.055251e-02
## V473 0.03924674 -5.639683e-02 -3.272833e-02
## V474 0.03839581 -5.504920e-02 -3.372862e-02
## V475 0.03747471 -5.314016e-02 -3.369209e-02
## V476 0.03815817 -5.829395e-03  5.592298e-02
## V477 0.03904920 -8.250844e-03  5.842826e-02
## V478 0.03982596 -1.069010e-02  6.014794e-02
## V479 0.04054603 -1.333070e-02  6.121580e-02
## V480 0.04127766 -1.635627e-02  6.085026e-02
## V481 0.04196566 -1.939139e-02  5.952146e-02
## V482 0.04244619 -2.283937e-02  5.734880e-02
## V483 0.04290800 -2.643639e-02  5.424121e-02
## V484 0.04323941 -2.994582e-02  5.016299e-02
## V485 0.04353966 -3.390187e-02  4.502339e-02
## V486 0.04369085 -3.760585e-02  3.937684e-02
## V487 0.04368534 -4.143224e-02  3.257160e-02
## V488 0.04373476 -4.463946e-02  2.569254e-02
## V489 0.04363517 -4.797083e-02  1.865080e-02
## V490 0.04342309 -5.086668e-02  1.147032e-02
## V491 0.04311297 -5.334562e-02  4.434193e-03
## V492 0.04266313 -5.563016e-02 -2.697567e-03
## V493 0.04211915 -5.713781e-02 -9.131312e-03
## V494 0.04157842 -5.818026e-02 -1.495745e-02
## V495 0.04097466 -5.876674e-02 -1.989568e-02
## V496 0.04031657 -5.881666e-02 -2.446056e-02
## V497 0.03953502 -5.835759e-02 -2.792362e-02
## V498 0.03867562 -5.723355e-02 -3.027041e-02
## V499 0.03784759 -5.553582e-02 -3.149893e-02
## V500 0.03695030 -5.331763e-02 -3.173705e-02
## V501 0.03783685 -1.100128e-02  5.551862e-02
## V502 0.03867317 -1.354315e-02  5.803530e-02
## V503 0.03941279 -1.608174e-02  5.977798e-02
## V504 0.04009756 -1.868861e-02  6.087569e-02
## V505 0.04076674 -2.165873e-02  6.057449e-02
## V506 0.04142121 -2.461266e-02  5.936610e-02
## V507 0.04185450 -2.793944e-02  5.724683e-02
## V508 0.04228028 -3.131119e-02  5.435173e-02
## V509 0.04257588 -3.455702e-02  5.046216e-02
## V510 0.04283344 -3.829900e-02  4.557272e-02
## V511 0.04294895 -4.165355e-02  4.019486e-02
## V512 0.04290839 -4.509054e-02  3.369235e-02
## V513 0.04295093 -4.788793e-02  2.710072e-02
## V514 0.04280904 -5.092986e-02  2.035227e-02
## V515 0.04257415 -5.337932e-02  1.342598e-02
## V516 0.04226500 -5.547073e-02  6.574251e-03
## V517 0.04178757 -5.732680e-02 -4.442497e-04
## V518 0.04124248 -5.842645e-02 -6.750048e-03
## V519 0.04071872 -5.912735e-02 -1.244989e-02
## V520 0.04010456 -5.944044e-02 -1.739703e-02
## V521 0.03945884 -5.910504e-02 -2.191491e-02
## V522 0.03871020 -5.838318e-02 -2.544520e-02
## V523 0.03787981 -5.695069e-02 -2.784916e-02
## V524 0.03709705 -5.507949e-02 -2.908730e-02
## V525 0.03624583 -5.276503e-02 -2.952382e-02
## V526 0.03748203 -1.542851e-02  5.403739e-02
## V527 0.03826384 -1.808384e-02  5.651066e-02
## V528 0.03896546 -2.065979e-02  5.824119e-02
## V529 0.03957558 -2.325380e-02  5.935447e-02
## V530 0.04019526 -2.608907e-02  5.910487e-02
## V531 0.04079232 -2.896469e-02  5.809282e-02
## V532 0.04118016 -3.207280e-02  5.616910e-02
## V533 0.04156204 -3.521075e-02  5.345867e-02
## V534 0.04181286 -3.812715e-02  4.973705e-02
## V535 0.04202850 -4.155937e-02  4.511859e-02
## V536 0.04210241 -4.461674e-02  4.005633e-02
## V537 0.04203559 -4.768419e-02  3.396076e-02
## V538 0.04205208 -5.008228e-02  2.781284e-02
## V539 0.04187295 -5.280034e-02  2.140802e-02
## V540 0.04162109 -5.489041e-02  1.481845e-02
## V541 0.04128401 -5.663086e-02  8.256410e-03
## V542 0.04081567 -5.808750e-02  1.508821e-03
## V543 0.04025138 -5.881883e-02 -4.712703e-03
## V544 0.03973592 -5.921593e-02 -1.021113e-02
## V545 0.03913309 -5.922900e-02 -1.502982e-02
## V546 0.03849433 -5.868210e-02 -1.949806e-02
## V547 0.03776810 -5.771093e-02 -2.300650e-02
## V548 0.03697893 -5.606319e-02 -2.545158e-02
## V549 0.03623595 -5.409331e-02 -2.678265e-02
## V550 0.03543872 -5.172943e-02 -2.731588e-02
## V551 0.03704713 -1.914277e-02  5.186687e-02
## V552 0.03776482 -2.180519e-02  5.421532e-02
## V553 0.03842215 -2.434806e-02  5.591794e-02
## V554 0.03897768 -2.687233e-02  5.695550e-02
## V555 0.03955537 -2.954415e-02  5.673980e-02
## V556 0.04011884 -3.221470e-02  5.586502e-02
## V557 0.04045043 -3.507843e-02  5.400690e-02
## V558 0.04078110 -3.791943e-02  5.149753e-02
## V559 0.04099668 -4.047410e-02  4.809561e-02
## V560 0.04115162 -4.357572e-02  4.382653e-02
## V561 0.04118965 -4.628594e-02  3.925270e-02
## V562 0.04106636 -4.891287e-02  3.362908e-02
## V563 0.04106138 -5.091790e-02  2.795712e-02
## V564 0.04086496 -5.323327e-02  2.193487e-02
## V565 0.04061867 -5.485914e-02  1.585790e-02
## V566 0.04028945 -5.628264e-02  9.688966e-03
## V567 0.03981561 -5.744268e-02  3.274650e-03
## V568 0.03927285 -5.789273e-02 -2.580121e-03
## V569 0.03878351 -5.810121e-02 -7.919913e-03
## V570 0.03822504 -5.790337e-02 -1.251411e-02
## V571 0.03761090 -5.722826e-02 -1.685589e-02
## V572 0.03692946 -5.615305e-02 -2.028802e-02
## V573 0.03620109 -5.438742e-02 -2.283772e-02
## V574 0.03549562 -5.242069e-02 -2.426762e-02
## V575 0.03474265 -5.012590e-02 -2.495190e-02
## V576 0.03669450 -2.255185e-02  4.905929e-02
## V577 0.03734134 -2.513727e-02  5.122647e-02
## V578 0.03794229 -2.754013e-02  5.273260e-02
## V579 0.03841033 -2.994522e-02  5.365861e-02
## V580 0.03890552 -3.235424e-02  5.351002e-02
## V581 0.03941150 -3.484249e-02  5.285896e-02
## V582 0.03966507 -3.740020e-02  5.115686e-02
## V583 0.03994815 -3.992744e-02  4.896663e-02
## V584 0.04011540 -4.213358e-02  4.596660e-02
## V585 0.04023203 -4.488716e-02  4.209830e-02
## V586 0.04023748 -4.712221e-02  3.802781e-02
## V587 0.04008994 -4.931723e-02  3.305902e-02
## V588 0.04008258 -5.090608e-02  2.797781e-02
## V589 0.03986870 -5.278262e-02  2.237407e-02
## V590 0.03962916 -5.396369e-02  1.691227e-02
## V591 0.03931379 -5.501805e-02  1.118159e-02
## V592 0.03884375 -5.589804e-02  5.160436e-03
## V593 0.03831473 -5.612504e-02 -2.734121e-04
## V594 0.03785213 -5.610422e-02 -5.334856e-03
## V595 0.03731162 -5.574342e-02 -9.718330e-03
## V596 0.03675690 -5.500915e-02 -1.381648e-02
## V597 0.03612302 -5.390760e-02 -1.711672e-02
## V598 0.03544140 -5.208959e-02 -1.979939e-02
## V599 0.03479047 -5.014645e-02 -2.140058e-02
## V600 0.03408342 -4.800313e-02 -2.223760e-02
## V601 0.03602191 -2.509035e-02  4.566008e-02
## V602 0.03658826 -2.749904e-02  4.765232e-02
## V603 0.03713057 -2.969826e-02  4.906954e-02
## V604 0.03754531 -3.182304e-02  4.996929e-02
## V605 0.03797016 -3.393971e-02  4.981761e-02
## V606 0.03843654 -3.622097e-02  4.932337e-02
## V607 0.03863926 -3.847801e-02  4.787223e-02
## V608 0.03887528 -4.053273e-02  4.598462e-02
## V609 0.03902424 -4.236439e-02  4.344353e-02
## V610 0.03910773 -4.477376e-02  4.006879e-02
## V611 0.03908338 -4.662635e-02  3.654121e-02
## V612 0.03891693 -4.841580e-02  3.213076e-02
## V613 0.03892134 -4.966559e-02  2.760142e-02
## V614 0.03869130 -5.123985e-02  2.255102e-02
## V615 0.03845639 -5.215519e-02  1.756306e-02
## V616 0.03816184 -5.292127e-02  1.231563e-02
## V617 0.03772393 -5.360463e-02  6.743542e-03
## V618 0.03721107 -5.367429e-02  1.760738e-03
## V619 0.03676649 -5.350668e-02 -2.990604e-03
## V620 0.03626011 -5.301675e-02 -7.143681e-03
## V621 0.03573300 -5.225609e-02 -1.099379e-02
## V622 0.03514289 -5.112592e-02 -1.421252e-02
## V623 0.03454458 -4.921269e-02 -1.690872e-02
## V624 0.03396976 -4.741736e-02 -1.854459e-02
## V625 0.03332419 -4.547791e-02 -1.958560e-02
grey_Palette <- colorRampPalette(c("black","grey"))


par(mfrow=c(1,3))
image(eigen_matrix_1, col = grey_Palette(256), ann = TRUE, axes = FALSE, main = "Eigenvector  1")
image(eigen_matrix_2, col = grey_Palette(256), ann = TRUE, axes = FALSE, main = "Eigenvector  2")
image(eigen_matrix_3, col = grey_Palette(256), ann = TRUE, axes = FALSE, main = "Eigenvector  3")

In summary

In this project, the MUSK data is studied in Task 1, and image (plane) data in Task 2. I mainly used PCA, and MDS analysis are used in TASK 1. No significant relationship is found between the feature vector and the results.

In TASK 2, for the image (plane) processing, PCA Analysisi is used. The image is sucessfully loaded. Worked on it and reconstructed as asked.

Note that the echo = FALSE parameter was added to the code chunk to prevent printing of the R code that generated the plot.